Topological classification and diagnosis in magnetically ordered electronic materials
Bingrui Peng, Yi Jiang, Zhong Fang, Hongming Weng, Chen Fang
TTopological classification and diagnosis in magnetically ordered electronic materials
Bingrui Peng,
1, 2, ∗ Yi Jiang,
1, 2, ∗ Zhong Fang, Hongming Weng,
1, 3 and Chen Fang
1, 3, 4, † Beijing National Laboratory for Condensed Matter Physics and Institute of Physics,Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China Kavli Institute for Theoretical Sciences, Chinese Academy of Sciences, Beijing 100190, China
We show that compositions of time-reversal and spatial symmetries, also known as the magnetic-space-group symmetries, protect topological invariants as well as surface states that are distinctfrom those of all preceding topological states. We obtain, by explicit and exhaustive construction,the topological classification of electronic band insulators that are magnetically ordered for each oneof the 1421 magnetic space groups in three dimensions. We have also computed the symmetry-basedindicators for each nontrivial class, and, by doing so, establish the complete mapping from symmetryrepresentations to topological invariants.
I. INTRODUCTION
Magnetic space groups (MSGs)[1] describe the sym-metry of lattices where spins are magnetically ordered.Magnetic ordering necessarily breaks time-reversal sym-metry, and, as the order parameter is a vector, usu-ally also breaks some point-group symmetries (rotationand reflection). In many magnetically ordered materi-als, specially those having antiferromagnetism, there area special type of composite symmetries: a group element m = g · T is the composition of a space-group (SG) sym-metry g and time-reversal symmetry T . Consider for ex-ample an anti-ferromagnetic Neel order polarized along z with propagation vector Q = ( π/a, , a isthe lattice constant. The lattice translation by one unitcell along x , { E | } , is broken, and time reversal, T , isalso broken, while their composition { E | } · T remainsa symmetry. (Here we use E to represent the identity3 × O (3).)According to the types of g in m = g · T , the MSGsare classified into four types: if g does not exist (so that m does not exist), the magnetic group is type-I; if g isidentity, { E | } , type-II; if g involves a nontrivial pointgroup operation, type-III; and if g is a pure lattice trans-lation, { E | klm } , the group is type-IV. By this definition,type-II MSGs contain time reversal, and hence describenonmagnetic materials. In this work, we focus on type-I,III, IV MSGs[2].MSGs are also the symmetries of the effective Hamilto-nians describing elementary excitations, such as magnonsand electrons, that move within a magnetically orderedlattice. In this work, we focus on magnetic materialsin which coherent quasiparticle fermion excitations formband(s) within a finite range of the Fermi energy, andstudy the band topology of these fermions. Our theorycan in principle be applied to any magnetic materials ∗ These authors contributed equally to this study. † [email protected] where the notion of electron-like quasiparticles is valid,at least near the Fermi energy. Itinerant magnets[3],heavy-fermion metals[4], and doped Mott insulators[5]that maintain a magnetically ordered state, are consid-ered to belong to this large class of materials.The interplay between symmetry and topology hasbeen a focus of modern condensed-matter research[6–9]. For a given symmetry group and a nonzero gap, allband Hamiltonians are grouped into equivalence classes ,where two Hamiltonians in the same (different) class(es)can(not) be smoothly deformed into each other, whilemaintaining both the gap and the symmetry. Each equiv-alence class is denoted by a unique set of integers calledthe topological invariants [10–17], the forms and types ofwhich only depend on the symmetry group and dimen-sionality. How many distinct equivalence classes existfor a given symmetry group in a given dimension, andwhat are the topological invariants for each class? Thisis the question we call the problem of topological classifi-cation . The theory of topological classification for time-reversal and particle-hole symmetries has been done in alldimensions using the K-theory[18–20]; the classificationproblem for a single spatial symmetry plus time rever-sal in three dimensions has been solved, usually heuris-tically, for several symmetries[21–31]; and the classifi-cation problem for arbitrary SGs plus time reversal inthree dimensions have been attempted using either the“real-space recipe” argument[30, 32, 33] or the “double-strong-topological-insulator construction”[27, 31] argu-ment more recently.The classification problem for MSGs begins with thetheory of axion insulators protected by space-inversionsymmetry without time reversal[7, 29, 34–40], followed bythe theory of antiferromagnetic topological insulators[41–44], again followed by the discovery of several topologi-cal invariants protected by wallpaper groups[45, 46]. Amore systematic attempt is made in Ref.[47], where thelayer-construction (LC) method reveals a number of newtopological states. In this work, we use the real-spacerecipe method developed in [32, 33]. Effectively, thismethod converts the problem of topological classification a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b into a “LEGO puzzle”, where one tries to find distinctways one can build an “edgeless” construction using somegiven pieces. This “LEGO puzzle” is then further trans-formed into finding all independent integer solutions ofa set of linear equations on a Z n ring. This method notonly yields a complete topological classification of gappedbands in each of the 1421 groups, but also gives us, foreach nontrivial equivalence class, one explicit and micro-scopic construction that we call the “topological crystal”(TC)[32]. We emphasize that our method finds moretopological classes than the layer-construction method,because LCs are a type of TCs, while not all TCs areLCs. (In fact, we show that at least 553 of the 1421MSGs have at least one topological state that cannot belayer-constructed.)If the classification gives us the labels (topological in-variants) for the equivalence classes into which gappedstates are put in, the topological diagnosis then tells us towhich equivalence class a specific, given material (Hamil-tonian) belongs. Ideally, a diagnosis scheme computesthe topological invariants of that Hamiltonian, and bycomparing these values with the labels on the equiva-lence classes, one puts the Hamiltonian into the rightone. However, topological invariants are notoriously dif-ficult to compute[24, 48], and for some, we do not evenhave the explicit expressions in terms of the wave func-tions of the bands[26, 27]. Fortunately, if we relax therequirement of ideal diagnosis to approximate diagnosis,the story is completely changed. An approximate diag-nosis uses partial information on the wave function, andin return gives us partial information on the topologi-cal invariants, not invariants themselves. For example,an approximate diagnosis for systems with n -fold rota-tion symmetry yields the invariant (Chern number) mod-ulo n by using only the rotation eigenvalues at severalhigh-symmetry momenta[29, 49]. Recently, the theory ofsymmetry-based indicators (SIs)[50] and that of topolog-ical quantum chemistry[51], enhanced by the full map-ping from indicators to topological invariants[30], givebirth to a fast approximate diagnosis scheme. This fast-diagnosis scheme has been applied to a large number ofnon-magnetic materials[52–54].In this work, we extend the above diagnosis schemeto magnetic materials that can be well-characterized interms of band structures. We find the explicit formu-las for all SIs in terms of band representations (which inpart differ from previous works[47, 55]); and we calcu-late the values of indicators for each TC in every MSG.Since each gapped state can be adiabatically continuedto a TC, this result in fact yields the mapping frominvariants to indicators. In this calculation, we findthat certain indicator values are never taken in any TC,and hence can only indicate nodal band structures [56–73]. These nodes are “evasive” as they are away fromany high-symmetry points or lines. In fact, we showthat all these nodal indicators indicate Weyl nodes atgeneric momenta. In a numerical diagnosis, the indi-cators are by far easier to obtain than the invariants, because the former only depend on the band represen-tations at fewer than or equal to eight momenta andthe latter depend on the valence-band wave functionsin the entire Brillouin zone (BZ). Therefore, an inversemapping from indicators to invariants/nodes is gener-ated using a script, provided along with the paper athttps://github.com/yjiang-iop/MSG mapping. II. CLASSIFICATIONA. General scheme
The basic idea of the real-space recipe is that all topo-logical crystalline insulators (TCIs) can be adiabaticallydeformed into a special form of real-space constructionscalled the TCs, such that classifying TCIs is equivalent toclassifying TCs. TCs are real-space patterns built fromtopological pieces in lower dimensions, which were firstapplied to non-magnetic SGs in order to obtain the fullclassifications of non-magnetic TCIs[32]. In this work, weapply the real-space recipe for the construction and theclassification of all gapped topological states protectedby MSGs.To start with, we build a structure of cell complexby using MSG symmetries, including both unitary andanti-unitary ones, and partition the 3D space into finite3D regions called asymmetric units (AUs) that fill thewhole 3D space without overlaps. In fact, all AUs aresymmetry-related and can be generated by choosing oneAU and then copying it using MSG operations. AUs arealso called 3-cells, and the 2D faces where they meet are2-cells. Similarly, 1D lines where 2-cells intersect are 1-cells, and the endpoints of 1-cells are 0-cells.To construct TCs in 3D, we should take account of all d -dimensional topological building blocks with d ≤
3. Foreach cell, its local symmetry group is defined as the col-lection of symmetries that keep every point of the cell un-changed. The local symmetry group of a cell determinesthe onsite symmetry class (Altland-Zirnbauer class[74])of the Hamiltonian on that cell. For MSGs, the effectivesymmetry class of a cell is always class A or class AI. Notethat although some cells have mirror plane as onsite sym-metry, the states on them can be divided into two sectorsby mirror eigenvalues, each of which belongs to class A.A 2-cell may have local symmetry group generated by M · T , and a 1-cell local symmetry group generated by C · T . Since ( M · T ) = ( C · T ) = +1, those cells be-long to symmetry class AI. According to the tenfold wayresults[20], systems of class AI have trivial classificationin all dimension ≤
3, and systems of class A have non-trivial classification in 2D and trivial classification in 0D,1D, and 3D, which means only 2D topological buildingblocks need to be considered in the construction of TCsfor MSG. Furthermore, there are two types of 2-cells de-pending on whether they coincide with mirror planes. Ifcoincide, they can be decorated with mirror Chern insu-lators characterized by two Z numbers, i.e., two mirrorChern numbers for ± i mirror sectors[75], and if not, withChern insulators characterized by one Z number, i.e., theChern number. Therefore, there are only two types ofbuilding blocks for our real-space recipe, i.e., Chern in-sulators and mirror Chern insulators.Having building blocks in hand, we next enumerateall topological inequivalent decorations on the cell com-plexes in MSGs. Before proceeding, note that the build-ing blocks themselves form a finitely generated Abeliangroup, e.g., Z for Chern insulators and Z for mirrorChern insulators. Therefore, the TCs built from themform a linear space with integer coefficients, such thattwo TCs can be “added” to obtain another TC, andthere exists a maximal set of linearly independent TCs(the generators) for each MSG. As a result, one justneeds to obtain the generators to describe the full setof TCs. As TCs are supposed to be fully gapped topo-logical states, all the boundary states contributed by 2Dbuilding blocks (Chern insulators and mirror Chern in-sulators) should cancel with each other on each 1-cell,leading to fully gapped states inside the bulk, a condi-tion known as the “gluing condition”[32] or “no-open-edge condition”[33]. After this procedure, we obtain a setof generators that form an Abelian group Z n . However,this is generally not the final classification, because somegenerators will reduce from Z -type to Z -type after aprocess of subtracting topological trivial elements called“bubble-equivalence”[32, 33]. The final classification canbe expressed as a quotient group Ker/Img, a structure re-sembling group (co)homology, where Ker stands for thelinear space of TCs satisfying no-open-edge condition andImg for the space of bubble-equivalence. These final clas-sifications of MSGs have the form Z n × Z l and can befound in Appendix.N.Lastly, we compare the real-space constructions fornon-magnetic SGs and MSGs. First, observe that theirbuilding blocks are different. In non-magnetic SGs, dueto the time-reversal symmetry (TRS), the building blocksare two-dimensional topological insulators and mirrorChern insulators with C + m = − C − m , while in MSGs, thebuilding blocks are Chern insulators and mirror Chern in-sulators with independent C + m and C − m [76]. More signifi-cantly, for non-magnetic SGs, one can obtain most of theTCs by LCs, which involve only layered 2D TIs and mir-ror Chern insulators as building blocks[30], with only 12non-magnetic SGs having states beyond LCs[32]. How-ever, non-layer constructions (non-LCs) exist widely inMSGs, and constructing with layers only may lose a num-ber of TCs. By contrast, the real-space recipe employingthe structure of the cell complex automatically includesboth LCs and non-LCs, giving the complete collection ofTCs, hence the complete classification of gapped topo-logical states. B. Topological invariants
TCIs are characterized by crystalline-symmetry-protected topological invariants. Previous works havelooked into some of the topological invariants protectedby magnetic crystalline symmetries, with the earliest onebeing the axion insulators with inversion invariant[29,36], followed by the anti-ferromagnetic topological insu-lators protected by anti-unitary translations[41, 42]. Inthis work, we exhaustively enumerate all topological in-variants in MSGs.Formally, having nontrivial symmetry-protected topo-logical invariants means a topological state cannot besmoothly deformed into a trivial state when the symme-try is preserved. Specially, if a TCI can not be adiabat-ically connected to an atomic insulator with the sym-metry operation g preserved, we say it has nontrivial g -invariant. As all TCIs are adiabatically connected toTCs, we can utilize TCs to derive all invariants protectedby MSGs symmetries. In our real-space recipe, given asymmetry operation g alone, if nontrivial TCs compati-ble with g can be constructed, then we say g can protectnontrivial topological invariants. More specifically, eachindependent TC corresponds to an independent invari-ant, and they have the same group structure, e.g., a Z TC owns a Z invariant.To find which symmetry operations in MSGs can pro-tect nontrivial invariants and what kinds of the invari-ants they protect, we take account of all of MSG symme-tries one by one, including translation, rotation, inver-sion, mirror, rotoinversion ( S n ), screw, glide, and thosecombined with TRS such as C n · T , etc. We considerwhen each of them is present alone, what TCs can beconstructed. If no TC exists, this symmetry has onlytrivial invariant; if there exist TCs, this symmetry mustprotect nontrivial topological invariants, and we furtherdecide how many independent TCs and whether theyare Z -type or Z n type by checking if the TCs can besmoothly deformed into trivial states after multiplying n times. In this way, we know all the MSG symmetriesthat can singly protect nontrivial topological invariants,with each symmetry hosting at most one invariant, beingeither Z -type or Z -type. We summarize the results inTable.I, and classify these invariants into three types: • Trivial invariants: C n rotations and anti-unitaryimproper point group symmetries in MSGs cannothost nontrivial decorations. Take C for example.When there is only a C symmetry alone, considera 2D plane that passes the C -axis, which is par-titioned into two 2-cells by the axis. Chern insu-lators cannot be decorated on the two 2-cells, be-cause their chiral edge modes on the C -axis are C -related, and as such are in the same directionand cannot gap out each other. This is differ-ent from the non-magnetic case where C has a Z invariant[30, 32], as the building blocks in non-magnetic SGs are 2D TIs and two C -related helical a i Ta i
1× 1× 1× |C m |× × t / (10) unitary screw {C n |t }, n=2,3 , n T, n=2,4,6(8) inversion (9) S (7) anti-unitary screw {C n |t}T, n=2,4,6(5) unitary glide (6) anti-unitary glide inversion/S center: rotation/screw axis: mirror/glide plane: t / chiral hinge mode:2D surface mode: Z× FIG. 1. Surface states of symmetries with non-trivial topological invariants in MSGs, with the surface terminations preservingcorresponding symmetries. These surface states can be 1D chiral hinge modes and 2D surface modes, with 2D modes beingeither slope-like chiral surface modes in (1),(6),(10), or Dirac cones in (2),(3),(4),(5). More details can be found in Appendix.C. edge states can cancel with each other. • Z invariants: Unlike non-magnetic MSGs whereonly mirror symmetries have Z invariant, here inMSGs, as the building blocks are both Z -type, wehave many other Z invariants, including the invari-ant of unitary translation, unitary screw, and anti-unitary glide. • Z invariants: A large proportion of MSG sym-metries protect Z invariants, with many of themunique to MSGs, such as anti-unitary transla-tions/rotations/screws. In Appendix.D, we definea special type of decoration called “ Z decoration”which has zero weak invariants and all Z invari-ants bond together and equal to 1 (for mirror Chernnumbers, we can define C + m + C − m mod 2 as a Z in-variant), and can be seen as an axion insulator withthe axion angle θ = π . For Z decorations, all these Z invariants merge into one Z invariant, i.e., the“axion invariant”, whose definition can be taken asthe 3D magnetoelectric polarization P accordingto Ref.[7, 34, 49].Each nontrivial invariant has its distinct anomaloussurface state due to the bulk-boundary correspondence.Because the topological invariants together with their MSG Symmetries Invariant type unitary rotation C n Trivialanti-unitary improper point groupsymmetries P · T , M · T , S n · T unitary improper S n , P , { M | } Z anti-unitary translation { E | t } · T anti-unitary proper C n · T , { C n | t } · T unitary translation { E | R } Z unitary mirror M unitary screw { C n | t } anti-unitary glide { M | } · T TABLE I. MSG symmetries and their corresponding invari-ant types. We use the Seitz Symbol { O | t } to represent sym-metries with nonzero translations, where R denotes a latticetranslation and t a fractional translation. The invariants ofthe unitary screw and anti-unitary glide are bound to thetranslation invariant, with their values being n and of thetranslation invariant, respectively. surface states can be superimposed, we only need to de-rive the surface state for every single invariant, and thesurface states of all TCIs can be readily known from theirinvariants. Among these surface states in MSGs, somehave already been discussed in previous works, includ-ing those protected by C · T [25, 48, 77], C · T [78],glide[48, 79–82], and anti-unitary translation[41, 42], etc.(for spinless invariants including C n · T and anti-unitarytranslation, see Ref.[43, 44]), while some are first pro-posed in this work, such as the one protected by anti-unitary glide. We plot these surface states in Fig.1, andmore details can be found in Appendix.C. As an exam-ple shown in Fig.1(3), on a cylinder geometry, the surfacestates protected by C n · T, n = 2 , , n chiral hinge modes relatedto each other under C n · T on the side surface.Although we find all the topological invariants pro-tected by MSG symmetries, we have not derived theirexplicit formulas in k-space, which could be complicated.Instead, we compute these topological invariants in real-space for the TCs we built using a unified method asfollows[32]. Given an MSG M and an element g ∈ M ,first choose a generic point r inside an arbitrary AU, thendraw a path connecting r and its image point g · r , underthe only constraint that the path does not cross any 1-cells and 0-cells. The corresponding invariant δ ( g ) is de-termined by the decorated 2-cells that the path crosses,i.e., the total Chern number or mirror Chern numberaccumulated through the path. The invariants thus cal-culated are well-defined and do not depend on the choiceof the generic point r or the path. The full listing of topo-logical invariants of the TCs will prove to be useful in thesecond part of this work, where we use SIs to diagnosetopological states, and TCs function as an intermediateto connect SIs and invariants.We take one example to show the correspondence be-tween invariants and surface states for a given MSG. Thetype-III MSG 3.3 P (cid:48) has three independent generatorsof TCs. One of them is protected by C y · T with nonzeroinvariant δ ( C y · T ) = 1, while the other two are pro-tected by the translation symmetries in x and z direc-tions, with nonzero weak invariant δ ( { E | } ) = 1 and δ { E | } = 1, respectively. For the first decoration, thesurface state protected by C y · T has been describedbefore, as shown in Fig.1(3), while the other two trans-lation decorations have one chiral surface mode on the2D surface preserving the translation symmetry in x/z direction, as shown in Fig.1(1). C. Non-layer constructions
The abundance of non-LCs distinguishes the TCs inMSGs from those in non-magnetic SGs, where most of thedecorations are LCs. In fact, LCs are just a special typeof TCs where 2D planes are uniformly decorated, whilenon-LCs contain non-uniform decorations or incomplete2D planes.We use type-1 MSG
P mmm as a representative toshow the characters of non-LCs.
P mmm has three or-thogonal mirrors M x , M y , and M z , which do not com-mute with each other due to the spin rotation. Placingmirror Chern insulator layers with ( C + m , C − m ) = (1 , − FIG. 2. The non-LC of
P mmm , where the arrows denote thedirections of the chiral edge modes on the 2-cells. Note weonly plot the 2-cells around the origin point inside the unitcell, and omit the six side-surfaces for simplicity, which arein fact all decorated. Assume the 8 inversion centers havecoordinate r = a δ + a δ + a δ , where δ i = 0 , a i is the lattice vector. The 2-cells around an inversion centerhave the same decoration as shown in the figure if (cid:80) i δ i iseven, while the 2-cells have opposite directional edge modesif (cid:80) i δ i is odd. on any of the six mirror planes, i.e., x, y, z = 0 , , forms 6independent LCs. However, one can still construct a dis-tinct non-layer decoration falling outside the linear spaceof these 6 LCs. As shown in Fig.2, this non-layer deco-ration is constructed by sewing small patches of mirrorChern insulators with ( C + m , C − m ) = (1 ,
0) or (0 , −
1) oneach mirror 2-cell, with adjacent patches having oppo-site mirror Chern numbers, making each mirror planenon-uniform. Each 1-cell is shared by 4 patches, withpatches in different directions contributing opposite chi-ral edge modes, canceling with each other and satisfy-ing the no-open-edge condition. This non-LC is distinctfrom LCs, where each mirror plane is decorated with auniform, infinite-sized mirror Chern insulator. The dec-orated 2-cells of this non-LC are all pinned on the mirrorplanes, preventing it to be deformed into an LC.Although this non-LC seems complicated, we observethat it can be connected to the time-reversal strongTI (STI) from the perspective of surface states. Onthe one hand, this non-LC has mirror Chern numbers( C + m,k i =0 , C − m,k i =0 , C + m,k i = π , C − m,k i = π ) = (1 , − , , , i = x, y, z , which lead to a nontrivial mirror-protected sur-face state with a single Dirac cone on k i = 0 in the 2Dsurface BZ. On the other hand, an STI put on P mmm lattice also has a single Dirac cone on each surface. Bythe principle of bulk-edge correspondence, we concludethat the STI with
P mmm symmetry can be adiabati-cally deformed into this non-LC by breaking the TRSwhile preserving crystalline symmetries. Note that for
P mmm , despite the absence of TRS, the three mirrorsanti-commuting with each other also enforce the energybands to form Kramer-like pairs, i.e., twofold degeneracywith the same parity and opposite mirror eigenvalues.Therefore, the three mirror symmetries still pin the sur-face Dirac cone on the mirror invariant lines when theTRS is broken. As a result, we can use the tight-bindingmodels of STIs, for example, the 3D Bernevig-Hughes-Zhang (BHZ) model[83], to describe this non-LC.As mentioned before, unlike non-magnetic SGs whereonly 12 SGs have non-LCs[32], non-LCs exist widely inMSGs. For instance, all the supergroups of MSG 47.249
P mmm can host a non-LC as one of its TCI classifica-tion generators, and similar for MSG 25.57
P mm
2, 84.51 P /m , 10.44 P (cid:48) /m , 75.3 P (cid:48) , 6.21 P a m , and 75.5 P C Z decoration”, are all axion insulators. AsSTIs are also axion insulators, when the TRS is broken,there still exist other unitary improper or anti-unitaryproper symmetries that have non-trivial Z invariants,preserving the quantized π axion angle. III. DIAGNOSIS
Symmetry-based indicator is a powerful tool for diag-nosing topological states[50, 51, 84] and has been appliedin both non-magnetic SGs[30, 31, 85] and MSGs[47, 55,86, 87], which leads to the discovery of a significant num-ber of new topological materials[52–54, 88]. SI theory inMSGs has been partly tackled in previous works, withthe SI group structures and part of the SI expressionsgiven.However, the mappings between SIs to topological in-variants in MSGs have not been fully investigated. Inthis work, we derive the explicit formulas of all indicators,and their quantitative mappings to topological invariantsif correspond to gapped states, and possible Weyl pointconfigurations if correspond to gapless states. These twodifferent correspondences to gapped and gapless statesare found thanks to the TCI classifications, which includeall the gapped TCI states, and states having nonzero SIwhile not belonging to any gapped state must be Weylsemimetals.
A. Explicit expression of indicators
Among 1421 MSGs, 688 of them have nontrivial SIs.Despite the seemingly large number, SIs in all MSGs canbe induced from 16 generating MSGs, which we list inTable.II, with only 10 corner cases to be discussed later.Most of the SIs can be expressed in terms of some topo-logical invariants. In fact, the idea of diagnosing non-trivial topology using only the symmetry data on high-symmetry points (HSPs) in the BZ stems from the Fu-
MSG X BS SI P Z , , , z P, , z P, , z P, , z P P mmm Z , , , z (cid:48) P, , z (cid:48) P, , z (cid:48) P, , z (cid:48) P P n, n = 2 , , , Z n z nC P n/m, n = 2 , , , Z n,n,n z + nm, , z − nm, , z + nm,π P Z , , z ,S , z , Weyl , z C P /mmm Z , , z (cid:48) P, , z +4 m,π , z P /mmm Z , z +6 m,π , z P nc (cid:48) c (cid:48) , n = 2 , , Z n z (cid:48) nC TABLE II. Generating SIs and generating MSGs in MSGs.We use a simplified notation to represent the SI group, i.e., Z n ,n , ··· = Z n × Z n × · · · . Among these generating MSGs,only P nc (cid:48) c (cid:48) are type-3 MSGs, while all the others are type-1.The definition and interpretation of these SIs can be found inAppendix.G. Kane formula[17] for non-magnetic SGs with inversionsymmetry, and in the following works[29, 36, 49] C n rota-tion eigenvalues are used to calculate the Chern numbermodulo n , i.e., the z nC indicator in Table.II. When thereexist mirror planes perpendicular to the rotation axis,the z nC indicator can be applied to two mirror sectorswhich give the z ± nm, /π indicator, where ± represents twomirror sectors. There are also three type-3 generatingMSGs P nc (cid:48) c (cid:48) , n = 2 , , C n eigen-values) on the k z = π plane, allowing the definition of anew set of SIs z (cid:48) nC using the number of degenerate pairs,the value of which corresponds to the Chern number di-vided by two modulo n of the 2D BZ. The interpretationsof other generating SIs are left in Appendix.G.The word “generating” means the SIs in other MSGscan be expressed in terms of these generating SIs. Sometimes, the “generation” involves the reduction of the in-dicator group, e.g. from Z to Z by taking only the evennumbers in Z . Moreover, the interpretation of the SI,generated from the same indicator, can be different in dif-ferent MSGs. For example, the z nC indicator originallyrepresents the Berry phase of a closed loop in the 2D BZas shown in Appendix.K, and it becomes the Chern num-ber when the state is gapped. However, as shown later,in some other groups, this indicator if nonzero indicatesgapless topological states, and its value represents thenumber of Weyl points. For another example, the z (cid:48) P indicator is adopted in some MSGs where the coirrepsare not doubly degenerate with the same parity. In thesecases, the interpretation of these indicators may changewhich takes case-by-case examination, and the SI expres-sions are only effective and may become invalid when, forexample, the origin point is changed. Thus we fix thecoordinate system to avoid these problems by adoptingthe Bilbao convention[89–91] when calculating the indi-cators.To sum up, the indicator formulas in generating SIscan be used to express all SIs in other MSGs, except10 corner-case MSGs which have SI formulas given inde-pendently using coirreps in Appendix.H. Here we showan example of MSG 42.222 F m (cid:48) m (cid:48)
2, which has a Z in-dicator with expression: z , . = N (Γ ) − N ( A ) mod 2 (1)where N ( ¯ K i ) represents the number of coirrep ¯ K i . ThisMSG has TCI classification Z , the generator of whichhas weak invariants δ w = (1 , ,
0) and all other invariantsequal zero. The SI z , . = 1 represents this generatoras well as all the odd number copies of it. Note that thisindicator is different from the choice in Ref.[47]. B. Computation of indicators for topologicalcrystals
In this section, we show how one can calcu-late the SIs for a given TC. As most of the SIscorrespond to some topological invariants, including z P,i , z P , z (cid:48) P , z nC , z ± nm, /π , z ,S , and z (cid:48) nC , their valuescan be determined directly. For example, the Z × Z × Z × Z indicators defined in P ¯1 have straightforwardmeanings: z P,i = δ w,i mod 2 , z P = 2 δ ( P ) (2)where δ w,i ( i = 1 , ,
3) is the weak invariant and δ ( P ) isthe inversion invariant. Note that z P = 1 , z (cid:48) P , z , and z .To determine their value for a TC, we need to find acompatible set of coirreps for the TC and then calculatethe SIs of the coirrep set. The result does not, however,depend on which particular set we choose as long as it iscompatible with the TC. This is because a given set ofinvariants, that is, a given TC, can only correspond toone possible set of SIs. Therefore, we only need to findone compatible set of coirreps for the TC and calculateits SI.Finding compatible coirreps for an LC is simple, but toa non-LC TC can be challenging. Fortunately, as shownbefore that many of the non-LCs in MSGs are adiabati-cally connected to STIs, the BHZ model describing STIscan be used, which has symmetries of SG 221 plus theTRS, a supergroup to many MSGs. As a result, we canadjust the model parameters such that the BHZ modeland the non-LCs share the same set of invariants, andthen compute the indicators for the BHZ model, withdetails in Appendix.L.The mappings between invariants and SIs only need tobe derived for the generating MSGs shown in Table.II.Other MSGs, except the corner cases and Weyl states,either are the supergroups of these generating MSGs orhave a different Bravais lattice, and their mappings canbe induced from the mappings of the generating MSGs. For corner cases, we derive their mappings case-by-casein Appendix.H. C. The mapping from indicators to invariants
So far we have obtained the mapping from invariants toindicators for topological gapped states protected by allMSGs, listed in Appendix.O. A practically more usefulmapping is its inverse, one from indicators to invariants.This is because, on one hand, the indicators only de-pend on the coirreps at HSPs, and as such are far easierto obtain in first-principles calculations. On the otherhand, the invariants, as shown in the previous section,are directly related to the topological surface states thatmay be detected experimentally. A mapping from indi-cator to invariants hence links first-principles calculationto experimental observables. These mappings from SIsto invariants are “one-to-many” in general, in which agiven SI set may be mapped to multiple invariant sets.Assume an MSG has m linearly independent TCs,which have invariant sets V , V , ..., V m and SI sets S , S , ..., S m . Given an arbitrary SI set S from theSI group, to find all the possible invariant sets corre-sponding to it, one needs to solve the linear equation (cid:80) mi =1 x i S i = S , where x i are coefficients to be deter-mined. The invariant sets corresponding to S can beobtained using the solved x i . This inhomogeneous linearequation can be solved by first finding a special solutionand then adding all the general solutions to its homoge-neous counterpart, i.e., the solutions of the correspond-ing homogeneous linear equation, (cid:80) mi =1 x i S i = 0, whichgives the invariant sets that have zero SIs.Following this line, we have developed an algorithmthat automatically computes all possible sets of invari-ants for a given set of indicators. The code and thefull results which have been entered into a large ta-ble may be downloaded from https://github.com/yjiang-iop/MSG mapping. D. Weyl semimetal indicators
Unlike non-magnetic spinful SGs[30] where all SIs in-dicate topological gapped states, we find many SIs inMSGs correspond to gapless Weyl semimetals. As thefull mapping from TCs to SIs has been found for eachMSG, indicator values that do not belong to the imageof this mapping necessarily correspond to gapless states.Although these Weyl semimetals have Weyl points atgeneric momenta in the BZ, the symmetries of MSGsrequire their creation or annihilation to happen at HSPs,which makes it possible to detect them using SIs. TheseWeyl semimetals can be classified into two types, withone being inter-plane Weyl points that lie between k i = 0and k i = π plane, and the other being in-plane Weylpoints that lie on k i = 0 or π plane, where i denotes themain rotation axis of the MSG. We leave the full dis-cussion of Weyl semimetals to Appendix.I, and use oneexample of type-4 MSG 81.37 P C k z = 0 /π plane, with Weyl points connected by S symmetry having opposite chirality, as shown in Fig.3.This MSG has TCI classification Z and SI group Z × Z .The SI group is larger than the TCI group. One of the Z indicator can be chosen as z ,S , the odd value ofwhich corresponds to the decoration with S invariant δ ( S ) = 1, while the other Z indicator can be chosen as z C /
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I. Introduction 1II. Classification 2A. General scheme 2B. Topological invariants 3C. Non-layer constructions 5III. Diagnosis 6A. Explicit expression of indicators 6B. Computation of indicators for topologicalcrystals 7C. The mapping from indicators to invariants 7D. Weyl semimetal indicators 7References 8Appendices 12A. Topological crystals, construction andclassification 121. Cell Complex 122. 2D building Blocks 123. No-open-edge condition 134. Bubble equivalence 145. Decorations, first encounter 146. Final TCI classification 15B. Topological invariants 17C. Surface states 20D. Decorations, second encounter 22E. Layer and non-layer constructions 24F. A brief review of the symmetry-based indicatortheory 25G. Generating SIs in MSGs 261. Generating SIs in type-1 MSG 26a. MSG 2.4 P X BS = Z , , , ,Classification= Z × Z P n, X BS = Z n , Classification= Z P n/m, X BS = Z n,n,n ,Classification= Z P , X BS = Z , , ,Classification= Z × Z P mmm, X BS = Z , , , ,Classification= Z P /mmm, X BS = Z , , ,Classification= Z
30 g. MSG 191.233 P /mmm , X BS = Z , ,Classification= Z P X BS = Z , ,Classification= Z × Z P nc (cid:48) c (cid:48) , X BS = Z n , Classification= Z z (cid:48) P indicator 33b. The BZ folding process 33H. Corner cases of SI 351. Corner cases in type-1 MSGs 352. Corner case in type-2 MSGs 353. Corner cases in type-3 MSGs 354. Corner cases in type-4 MSGs 365. Interpretation of the type-3 corner caseindicator formulas 36I. Weyl semimetals in MSGs 401. Type-1 MSGs 402. Type-4 MSGs 403. Type-3 MSGs 41J. Summary of generating SIs in MSGs 431. Generating SIs in type-1 & 2 MSGs 432. Generating SIs in type-3 MSG 443. Corner cases SI 44K. The Berry phase of closed loops in 2D BZ 45L. The BHZ model for describing non-layerconstructions 461. 2D Chern insulator model 462. 3D BHZ model 47M. The construction of atomic insulator basis inMSGs 501. General procedures 502. Modification of the procedures foranti-unitary symmetries 51N. Table: TCI classifications of MSGs 52O. Table: Invariants and SIs of TCI classificationgenerators for MSGs with non-trivial SI group 561. Notations in the table of mapping 56P. Table: Invariants of TCI classificationgenerators for MSGs with trivial SI group 1082 Appendices
Appendix A: Topological crystals, construction andclassification
In this section, we discuss in detail the complete proce-dure of constructing topological crystals (TCs) protectedby magnetic space group (MSG) symmetries using ourreal-space recipe, including cell-complex, building blocks,no-open-edge condition, and bubble equivalence.
1. Cell Complex
To construct a TC, one first needs to find the geometricstructure called cell-complex, which is obtained by usingthe elements of crystalline symmetry group to cut the3D space into small regions and lower-dimensional pieces,i.e., 3, 2, 1, and 0-dimensional cells: •
3D regions are called 3-cells. A single 3-cell isalso called an asymmetric unit (AU), which fillsthe whole space without overlap after copied by allsymmetry group operations; •
2D planes where two 3-cells meet are called 2-cells; •
1D lines where 2-cells intersect are called 1-cells; •
0D end-points of 1-cells are called 0-cells.One thing worth noting is that, for any d-cell with d = 0 , , ,
3, two different points inside it cannot be re-lated by any crystalline symmetries, which means a sin-gle d-cell owns no crystalline symmetries itself. However,some symmetries can keep every point of the d-cell invari-ant, forming its on-site symmetry group. The propertythat a single d-cell having no crystalline symmetries issignificant, as one can only place topological states pro-tected by onsite symmetries on a d-cell, whose classifica-tions are already known from the ten-fold way results[20].The cell complex for MSGs is similar to that for spacegroups (SGs) discussed comprehensively in Ref.[32], withthe only difference being that MSGs can have anti-unitary symmetries, including magnetic half translations,which makes the cell complex more complicated.We show in Fig.4 the cell-complex of type-1 MSG 75.1 P
4. The unit cell is divided into four cuboids by C symmetry. The AU, or the symmetry-independent 3-cell,could be chosen as any one of the four cuboids. We choosethe 3 colored faces as symmetry-independent 2-cells andthe 5 blue lines as symmetry-independent 1-cells.
2. 2D building Blocks
Next, we consider what topological states can be dec-orated on the cell complex. To begin with, we have O C
123 4 5
12 3
FIG. 4. The cell complex of P
4, where the red number 1-3mark three 2-cells and blue number 1-5 mark five 1-cells. to decide the effective symmetry classes, i.e., Altland-Zirnbauer classes[74], of d-cells in the cell-complex, whichare determined by their on-site symmetry groups. Wenotice that for all d-cells with d = 0 , , , g , the states onthem can be diagonalized according to the eigenvalues of g into different sectors, with each sector belonging to classA. Some d-cells belong to class AI since they have onsiteanti-unitary symmetries with square +1, e.g., the com-bination of mirror and time-reversal symmetry (TRS), M · T . Thanks to the well-know result of ten-fold wayclassifications, the classification of class A is trivial when d = 0 , , Z when d = 2, while class AI have triv-ial classifications for d ≤
3. Therefore, we only need toconsider nontrivial decorations on 2-cells, which can befurther divided into two cases:1. When the 2-cell does not coincide with any mirrorplanes, the effective symmetry class is just classA with Z classification in 2D and the topologicalstate to be decorated on it is nothing but the Cherninsulator.2. When the 2-cell coincides with a mirror plane, wecan divide the states into two mirror ± i sectors,with each sector’s effective symmetry class beingA and having Z classification, which correspondsto the mirror Chern number[22, 23]. In this case,the topological state decorated on it is the mirrorChern insulator. Note that here TRS is absent, sothe two mirror Chern numbers C + m and C − m are notnecessarily opposite, and the total Chern number C + m + C − m are generically nonzero.In summary, there are two kinds of 2D building blocksfor constructing TCs protected by MSG symmetries: (i)Chern insulators and (ii) mirror Chern insulators.3
3. No-open-edge condition
Given a certain MSG, finding all independent mag-netic TCs is equivalent to enumerating all the topolog-ically distinct ways of decorating the cell complex. Astopological states we construct are fully gapped in thebulk, all the boundary states contributed by the 2-cellsmust cancel with each other on the 1-cells they meet, acondition known as the “non-open-edge condition”.We discuss the no-open-edge condition for the twotypes of building blocks separately:1. For non-mirror 2-cells, the no-open-edge conditionrequires an equal number of opposite directional 1Dedge modes such that they cancel with each otherand result in a fully gapped state on the 1-cell.Below we show an example of a 2D non-mirrorplane sliced by C rotation into two 2-cells. Asshown in Fig.5(a), when the 2-cells are parallel tothe C axis, the C -related edge modes run in thesame direction and cannot be canceled. However,when the 2-cells are vertical to the C axis shown inFig.5(b), the edge modes on the intersecting 1-cellhave opposite directions and can be canceled. Theno-open-edge condition fails for the first case whileholds for the second case.2. For mirror 2-cells, the no-open-edge condition ismore complicated because not only the chiralitiesbut also the mirror eigenvalues of the edge modesneed to be considered, i.e., the total chiralities ofedge modes should be zero for both mirror ± i sec-tors. In another world, two edge modes can begapped out if and only if they have the same mir-ror eigenvalue and opposite chiralities.For example, see a mirror plane passes the C or C · T axis, as shown in Fig.5(c)(d). For the C case,note C preserves the direction of edge mode andinverts the mirror sector, as the C and mirror axesare perpendicular which leads to C M = − M C .As a result, the two edge states on the 1-cell haveopposite mirror eigenvalues and the same chirality,which can not be gapped out.For the C · T case, note that T also inverts mir-ror sector as [ M, T ] = 0 and T ( ± i ) = ∓ i . Thecombination of C and T keeps the mirror sectorunchanged. T also inverts the chirality. Thus thetwo edge modes on the 1-cell have the same mir-ror eigenvalue but opposite chiralities, which canbe gapped out.Practically, we define a “boundary map” which mapssymmetry-independent 2-cells to symmetry-independent1-cells. This boundary map decides the way how bound-ary states are contributed on each 1-cell by all 2-cellsattached to it, written in a matrix form of dimension N × N . By calculating the kernel space of this C C (a) (b) C=1 C=-1 C=1 C=1C (c) C m =(1,0) C m =(0,-1) C T (d) C m =(1,0) C m =(1,0) FIG. 5. No-open-edge conditions. (a). Two C -related 2-cells that pass the C axis fail the no-open-edge condition.(b). Two C -related 2-cells vertical to the C axis satisfythe no-open-edge condition. (c). A mirror plane decoratedwith mirror Chern insulators that passes the C axis fails theno-open-edge condition. (d) A mirror plane decorated withmirror Chern insulators that passes the C · T axis satisfiesthe no-open-edge condition. matrix, one obtains the generators of decorations thatsatisfy the no-open-edge condition.We use the example of MSG 75.1 P P c , c , c andfive independent 1-cells labeled as b , b , b , b , b . Thematrix of boundary map is calculated as −
20 0 4 (A1)where the i -th rows represent 1-cell b i and the j -th col-umn represent 2-cell c j . The ( i, j )-th entry tells us howmany edge modes will be contributed on the i -th 1-cellif one puts a unit decoration, i.e., a Chern insulatorwith Chern number C = 1, on the j -th 2-cell. The ker-nel space of this matrix is spanned by only one vector,(1 , , , ,
0) means only the 2-cell c can be decoratedwith Chern insulators, and after copying c using C andtranslation symmetries, this decoration turns out to be4a 3D quantum anomalous Hall insulator (QAHI), i.e.,Chern insulators stacked along z direction.
4. Bubble equivalence
By solving the no-open-edge condition for MSGs, onegets the generators of TCs forming a Z n group. How-ever, the states generated by superimposing these gen-erators are not all necessarily nontrivial, and one needsto consider the so-called “bubble equivalence” processto remove the trivial states and arrive at the finial TCIclassification. Sometimes an even number of copies of agenerator could be trivialized, resulting in a Z classifi-cation.Specifically, a “bubble” represents a topological triv-ial state in 3D space which has a 2D surface with non-vanishing Chern number, as shown in Fig.6(a). It canbe created at a generic point inside an AU and growlarger until it coincides with the AU. Despite its surfacewith nonzero Chern number, a bubble itself is intrinsi-cally trivial since it can smoothly shrink back into a pointwithout breaking any symmetries.When a bubble is created in an AU, there will besymmetry-related bubbles created simultaneously insideall the symmetry-related 3-cells, and the relative signs ofsurface Chern numbers of these bubbles are also deter-mined by symmetries. On the 2-cell where two bubblesmeet, the Chern number can be changed by either ± ± ±
2, the TCI classifica-tion is affected by bubble equivalence.For example, we consider the type-4 MSG 1.3, P S z direction aside from three unitary integer transla-tions. Note that the layer generated by anti-unitaryhalf translation has the opposite Chern numberfrom the original layer. As shown in Fig.6(b), thedecoration with even Chern numbers on each layercan be trivialized by bubble equivalence, whichleads to the Z TCI classification of MSG 1.3,whose generator can be chosen as a layer construc-tion with C = 1 on integer planes and C = − C + m , C − m ) = ± (1 ,
1) on the 2-cell where they meet. This property is significantand will be used frequently in the following context.Without loss of generality, we set the mirror in z direction. Note the Berry curvature defined on x-yplane, ( ∇ × (cid:126)A ) z = ∂ x A y − ∂ y A x , is a pseudo vectorwhich is unchanged under M z . Thus two mirror-symmetric bubbles always modify the total Chernnumber on the mirror plane by an even number,i.e., the M z -related surfaces of two bubbles havethe same Chern number.Next, we decide the mirror eigenvalues of two M z -related bubbles. Denoting the states on the twobubbles as ψ and ψ . They transform to eachother under M z as: ψ → ζψ , ψ → ζ (cid:48) ψ , where ζ · ζ (cid:48) = − M z = −
1. Thus the matrix rep-resentation of M z is (cid:18) ζζ (cid:48) (cid:19) , which has two eigen-values ± i . One can construct two linear combina-tions of ψ and ψ , i.e., √ | ζ | ( iψ + ζψ ) and √ | ζ (cid:48) | ( ζ (cid:48) ψ − iψ ), as the two eigenstates withmirror eigenvalue + i and − i , respectively. As aresult, a minimal bubble equivalence changes themirror Chern numbers by ( C + m , C − m ) = ± (1 ,
5. Decorations, first encounter
According to previous discussions, there are two typesof building blocks, i.e., Chern insulators and mirrorChern insulators, according to their coincidence withmirror planes or not. Nevertheless, we can also classifythe building blocks in another way for the convenienceof the further procedure, that is, whether the net Chernnumbers they host are zero or not. The decorations withzero net Chern number are called M-building blocks, andif nonzero, C-building blocks.M-building blocks are only decorated on mirror 2-cellsand are actually mirror Chern insulators with zero netChern numbers, i.e., C + m + C − m = 0. By contrast, C-building blocks can be decorated on both mirror and non-mirror 2-cells, in both cases with nonzero net Chern num-bers. In another word, C-building blocks are either Cherninsulators or mirror Chern insulators with C + m (cid:54) = − C − m . a. Decorations by M-building blocks In practice, forMSGs with mirror symmetries, one does not have to dealwith no-open-edge condition for two mirror sectors di-rectly, which could be technically complicated as somesymmetry operations can invert the direction of a mirrorplane thus convert mirror + i sector to mirror − i sectoror vise versa.As discussed in Ref.[32], when decorated with M-building blocks, all 2-cells on the mirror plane can al-ways be “glued” together and extend to the whole mir-ror plane, which means the non-open-edge condition does5 BubbleBubble C m =(1,1) C=-1C=-1C=1C=1
C=-1C=-1BubbleC=-1C=1 C=-2C=-2C=2
BubbleC=-1C=1 BubbleC=-1C=1 + C=0C=0C=0 (a) (b) (c) T t / t Mirror plane Mirror plane
FIG. 6. Bubble equivalence. (a). A simple bubble with surfaces Chern number C = ±
1, where we only plot the upper andlower surfaces, and the side surfaces are in fact all have ± C = ± C + m , C − m ) = ± (1 , not need to be considered explicitly, as well as the bubbleequivalence which only effects decorations by C-buildingblocks. Instead, the number of independent decorationsis obtained by enumerating all symmetry-independentmirror planes. This step, i.e., decorating M-buildingblocks, results in a Z m group as part of the final clas-sification, where m is the number of independent mirrorplanes.In a word, decorations by M-building blocks can beseen as the whole collection of decorations with zero netChern numbers on all 2-cells. b. Decorations by C-building blocks To arrive at thefinal classification, the next step is to find the comple-ment of decorations by M-building blocks, i.e., all thedecorations with (at least some) 2-cells hosting nonzeronet Chern numbers, which are exactly the decorations byC-building blocks.In this step, even for mirror 2-cells, we only have toassign one Chern number, i.e., the total Chern number.The reason is as following. Note we already have thedecorations by M-building blocks, which can be freelyadded or subtracted now as they can be regarded as triv-ial elements in the space of decorations by C-buildingblocks. More specifically, if we assign two mirror Chernnumbers, ( C + m , C − m ), on a mirror 2-cell, we can add an M-building block with ( C − m , − C − m ) to it, resulting in a stateof ( C + m + C − m , Z l × Z n by decoratingC-building blocks, where n = 0 ,
1, which means there isat most one Z generator and will be explained later.
6. Final TCI classification
Finally, we combine the results obtained by decorat-ing M-building blocks and C-building blocks into Z m × Z l × Z n . However, this could still not be the final clas-sification, because when there are mirror planes, the Z generator obtained in the second step can be absorbedinto Z m and results in Z m × Z l . This is a simple caseof group extension that, after doubling and a process ofbubble equivalence, the Z generator becomes a decora-tion by M-building blocks with each mirror plane having C + m = − C − m , i.e., an element in Z m , which will be ex-plained more specifically later.Among our final results , we remark two points aboutthe TCI classifications in type-4 MSGs are worth men-tioning: • when the MSG has no mirror symmetry, the clas-sification is always Z ; • when the MSG has mirror symmetries, the clas-sification is always Z m , since the Z generator isabsorbed into mirror decorations, i.e., decorationsby M-building blocks.This means that the Z generator always exists for type-4MSGs, i.e., no type-4 MSG has trivial classification. Theabove two conclusions, although not proved rigorously,could be understood from the following arguments: • Z l generators (which correspond to translation Z decorations, as discussed below) are prohibited byanti-unitary translations in type-4 MSGs; • The simplest type-4 MSG P S
1, which has anti-unitary half-integer translation in one direction andunitary integer translations in other two directions,has a Z generator of TCI classification (corre-sponding to a Z decoration, as discussed below)which has a nonzero invariant of the anti-unitaryhalf translation. For an arbitrary type-4 MSG, its Z generator can be seen as a complication of the Z generator in P S
1, as other crystalline symme-6tries will add more decorated 2-cells, but cannottrivialize the original Z decoration in P S Appendix B: Topological invariants
Topological invariants quantitatively distinguish TCIsfrom trivial insulators. Formally, if a state can not be adi-abatically deformed into a trivial state preserving sym-metry g , it has a nontrivial topological invariant pro-tected by g .To find all MSG symmetries that can independentlyprotect topological invariants, we consider when a sin-gle generator of MSG symmetry operation g is present,what TC can be constructed. More specifically, we firstconstruct the cell complex with respect to symmetry g ,and then decorate possible 2D building blocks on 2-cellsunder the no-open-edge condition. Lastly, we check theeffect of bubble equivalence on the decorations. We say g has (i) trivial invariant if no TC can be constructed, (ii) Z invariant if there exists a non-trivial TC and can not betrivialized by copying arbitrary times, or (iii) Z n invari-ant if there exists a non-trivial TC and can be trivializedby copying n times. MSG Symmetries Invariant type unitary rotation C n Trivialanti-unitary improper point groupsymmetries P · T , M · T , S n · T unitary improper S n , P , { M | } Z anti-unitary translation { E | t } · T anti-unitary proper C n · T , { C n | t } · T unitary translation { E | R } Z unitary mirror M unitary screw { C n | t } anti-unitary glide { M | } · T TABLE III. MSG symmetries and their corresponding in-variant types. The invariants of the unitary screw and anti-unitary glide are bound to the translation invariant.
We enumerate all topological invariants protected byMSG symmetries in Table.III, which are either Z -typeor Z -type, and their corresponding minimal nontrivialdecoration, i.e., with invariant δ ( g ) = 1, in Fig.7.1. Unitary integer translation. Its corresponding in-variant is nothing but the weak invariant δ w,i . Aminimal decoration with δ w,i = 1 is a simple LCby placing C = 1 layers on integer planes in the a i direction. After doubling n times, this LC is stillnontrivial with layers having C = n , which con-firms the Z -type of weak invariants.2. Anti-unitary translation. A minimal decorationwith δ ( { E | a i } · T ) = 1 is a LC with C = 1 layerson integer planes and C = − a i direction. Its doubled state can betrivialized by the bubble equivalence, confirming its Z -type.3. Anti-unitary rotations. As shown in Fig.7(3), for C · T and C · T , their minimal decorations have 1 and 3 layers passing the rotation axis, respectively,while for C · T , there are 4 half-layers that intersectat the rotation axis, forming a non-LC. Their dou-bled states can all be trivialized, validating their Z -type.4. Mirror. With only mirror symmetry, the 3D spaceis separated by a single mirror plane (the only 2-cell) into two semi-infinite regions (the two 3-cells),and we need to consider decorations on the mir-ror plane and the effect of 3D bubbles. With-out TRS, the two real-space mirror Chern numbers( C + m , C − m ) on the mirror plane can take independentvalues, and they seem to serve as two generatorsof decorations, i.e., ( C + m , C − m ) = (1 ,
0) and (0 , Z × Z group. However, as discussed inSec.A 4, a decoration with ( C + m , C − m ) = ± (1 ,
1) canbe trivialized by the bubble equivalence. There-fore, there is only one independent generator, whichcan be chosen as either (1 ,
0) or (0 , Z × Z / { z = z } = Z group. We plot a minimaldecoration of mirror by placing a ( C + m , C − m ) = (1 , C diff = C + m − C − m and C total = C + m + C − m . Notethe bubble equivalence has no effect on C diff , butreduces C total from Z -valued to Z -valued, so thetwo redefined mirror Chern numbers seem to form a Z × Z group. However, the two generators of Z and Z are not independent, because the Z generatorcan also be the generator of Z , thus can be absorbedinto Z , leading to a Z group as the final result.5. Unitary glide. A minimal decoration with δ ( { M | a i } ) = 1 is shown in Fig.7(5), which hastwo layers perpendicular to the glide plane with C = ± { M | a i } ) = { E | a i } ,where a i is a lattice vector inside the glide plane,the existence of this non-symmorphic glide symme-try indicates the presence of translation symmetry a i . This decoration is similar to the anti-unitarytranslation decoration, which confirms the Z -typeof the glide invariant. We remark that this LC ofhorizontal layers can be deformed into another LCof vertical layers preserving the glide symmetry.6. Anti-unitary glide. Its minimal decoration has asimilar configuration with the unitary glide, butthe Chern number of the two layers in the unit cellare both C = 1. As ( { M | a i } · T ) = { E | a i } ,the existence of the anti-unitary glide also im-plies the translation symmetry, with their invari-ants bound together and both being Z -type, i.e., δ ( { M | a i } · T ) = δ w,i , which enforces δ w,i totake even numbers. For this minimal decoration,we have δ w,i = 2 and δ ( { M | a i } · T ) = 1.7. Anti-unitary screws. The minimal decorations of C n · T screws are similar to those of C n · T rotations.8 C=1C=-1 a i Ta i t / t / t / t / (1) unitary translation (2) anti-unitary translation(4) mirror(8) inversion (9) S (5) unitary glide (6) anti-unitary glide t / C=1C=1 C=1C=1C=-1C=1 C=1 C=1 C=1 C=1C m =(1,0) C=1 t / C=1 (10) unitary screw {C n |t }, n=2,3 , n T, n=2,4,6(7) anti-unitary screw {C n |t}T, n=2,4,6 FIG. 7. The minimal decoration of the corresponding topological invariant.
C=1C=-1 (a) (c) (b)
C=1 C=-1
FIG. 8. (a)-(c). The process of deforming horizontal layersinto vertical layers preserving the glide symmetry.
Moreover, these decorations can be transformedinto layers vertical to the screw axis with alter-nating Chern numbers preserving the anti-unitary screw symmetry, as shown in Fig.9. We plot thetransformation process for { C | a }· T in Fig.9(4).These invariants are also Z -type, as their doubleddecorations can be trivialized. C=-1 t / t / t / C=-1C=1 C=1 C=-1C=1C=-1C=1 (1) {C |1/2}T (2) {C |1/4}T (3) {C |1/6}T(4) FIG. 9. (1)-(3). Minimal decorations of anti-unitary screws { C n | n a i } · T, n = 2 , ,
6, which are equivalent to those inFig.7(7). (4). The process of deforming two horizontal layersinto vertical layers, preserving the { C | a } · T symmetry.
8. Inversion. A minimal decoration with δ ( P ) = 1 isconstructed by placing a C = 1 layer that passes9the inversion center. Its doubled state, i.e., a C = 2layer, can be trivialized by the bubble equivalence,or equivalently, split into two C = 1 layers andmove to infinity in an inversion-symmetric way.Thus the doubled state has δ ( P ) = 0 and confirmsthe Z -type of δ ( P ).9. S . A minimal decoration with δ ( S ) = 1 can bea non-LC with four half-layers similar to the deco-ration of C · T , as shown in Fig.7(9). But unlike C · T , this non-LC can be transformed into a singlelayer that passes the S -center preserving S sym-metry. δ ( S ) is Z -type as its doubled state can betrivialized.Note there are four rotoinverisons S n , n = 2 , , , S = M, S = C · P, S = C · M , and S is the only independent symmetry in the sensethat it cannot be decompose into a direct productof smaller point groups. As a result, we only discussthe minimal decoration S .10. Unitary screws. In Fig.7(10), we plot minimal dec-orations of { C n | n a i } , which have n layers with C = 1 connected by the screw in a unit cell, andcan be seen as a n -time copy of the translation dec-oration.In general, a unitary screw can be { C n | ml a i } , where l, m are coprime numbers, e.g., and . In thesecases, there are also l layers in a unit cell, whichhas δ w,i = l and δ ( { C n | ml a i } ) = m .As discussed in the main text, these topological invari-ants can be calculated for TCs in an intuitive way. Fora given symmetry g , first, choose a generic point r insidean arbitrary AU, and then use g to transform it to its im-age point g · r . Draw a path connecting these two pointswithout touching any 1- or 0-cells. The invariant δ ( g ) isdetermined by the decorated 2-cells through the path: • For every symmetry g , δ ( g ) can be the net Chernnumber accumulated through the path. Even if g is a mirror symmetry, we can ignore the two mirrorsectors and count only the net (total) Chern num-ber. We call this type of invariant defined by thenet Chern number “C-invariants” and denote themas δ C ( g ). • For a mirror symmetry M , we can also considerthe Chern numbers for each mirror sectors, i.e., thereal-space mirror Chern numbers ( C + m , C − m ), whichare obtained by counting the mirror Chern insula-tors decorated on the mirror plane. The momen-tum space mirror Chern numbers can be calculatedfrom the real space mirror Chern numbers, withprocedure similar to that for non-magnetic caseshown in Ref.[30].From the above algorithm for calculating C-invariantsfor TCs, a homomorphism between the symmetry oper-ations and their invariants naturally arises, as the Chern number accumulated through the path can be superim-posed: δ C ( g ) + δ C ( g ) = δ C ( g · g ) (B1)With this homomorphism, one only needs to calculatethe C-invariants for the generators of an MSG in orderto obtain all the invariants. However, we remark that assome C-invariants are Z -type while others are Z -type,it is the original value calculated from the path r → g · r before taking module that should be adopted whenapplying this homomorphism.In the following text, we sometimes use “invariants” toindicate “C-invariants” for simplicity, since in most casesthey are completely the same except for mirrors. Thespecific meaning of “invariants” can be inferred from thecontext.Interestingly, we notice the types of C-invariants areclosely related to the types of corresponding symme-try operations, especially for those protected by mag-netic point group (MPG) symmetries. As can beenfrom Table.III, proper unitary symmetries have triv-ial C-invariants, while anti-unitary proper symmetrieshave Z C-invariants (here mirror is special, as strictlyspeaking its invariant should be Z -type, neverthelesswe could regard the total Chern number as its Z C-invariant); unitary improper symmetries have Z C-invariants, while anti-unitary improper symmetries havetrivial C-invariants.0
Appendix C: Surface states
Surface states are in one-to-one correspondence withthe topological invariants. In other words, a nontrivial in-variant has a corresponding nontrivial surface state, anddifferent surface states can be superimposed. We enumer-ate all surface states protected by topological invariantsin MSGs, which can be seen from their minimal decora-tions. Note when constructing the minimal decorationsof invariants in Fig.7, we ignore the three lattice transla-tions. However, when constructing the surface states ofthese invariants, we sometimes need to restore the latticetranslations and require irrelevant invariants to be zero.1. Unitary integer translation { E | a i =1 , , } . The cor-responding surface state is known as the 3D quan-tum anomalous Hall effect. There are Z = δ w,i chiral surface modes on a i -preserving planes, withthe surface modes look like a slope in the 2D surfaceBrillouin zone (BZ).2. Anti-unitary translation { E | a i =1 , , } · T . As dis-cussed in Ref.[41, 42], there are an odd number ofDirac cones on a i -preserving planes, where a i is thedirection of anti-unitary translation. These Diraccones can only appear at the 4 TRIMs in the 2Dsurface BZ.3. Anti-unitary rotation C n · T, n = 2 , ,
6. Surfacestates protected by C · T and C · T have beenpreviously discussed in Ref.[25, 48, 77] and Ref.[78],respectively.In general, the surface states for C n · T, n = 2 , , C · T , the position of the Diraccone is unpinned[48], while for C n =4 , · T , the Diraccone is pinned at Γ (or a C n =4 , · T -invariant point)in the surface BZ. Note for C · T , the single Diraccone can split into three C · T -symmetric Diraccones by adding trivial bands.These surface Dirac cones can be analyzed from thesurface Hamiltonian as follows. Consider a minimalDirac surface Hamiltonian H ( k ) = v x k x σ + v y k y σ (C1)which is invariant under C n · T symmetry, i.e.,( C n · T ) H ( k )( C n · T ) − = H (( C n · T ) − k ) (C2)where C n = e i πn σ , T = iσ K (C3)and σ i,i =1 , , are the three Pauli matrices. Massterm proportional to σ is forbidden by C n · T , i.e.,one can not add mass term to gap the Hamiltonian H ( k ). However, the doubled Hamiltonian τ ⊗ H ( k ) can be gapped by adding mass term proportionalto τ ⊗ σ , which means a single Dirac cone is stablewhile two Dirac cones can be gapped. Therefore,the classification of this surface state is Z , in ac-cordance with the Z invariant.4. Mirror. As shown in Fig.10(4), when a mirror-preserving termination is made, the mirror-invariant planes in the 3D BZ now become mirror-invariant lines in the 2D surface BZ. Surface statesprotected by mirror symmetries are determined bythe momentum space mirror Chern numbers oneach mirror-invariant lines in the surface BZ. In thefollowing, we discuss the surface states on a spe-cific mirror-invariant line of the surface BZ, with( C + m , C − m ) denoting the momentum space mirrorChern numbers on it. If C ± m >
0, there are C ± m right-moving surface modes with mirror eigenvalue ± i , while for C ± m < | C ± m | left-movingmodes with mirror eigenvalue ± i (the direction ofsurface modes is convention-dependent and can beexchanged). As a result,(a) when C + m = − C − m , there are | C m | = | C + m − C − m | Dirac cones, as plotted in Fig.10(4).(b) when C + m (cid:54) = − C − m : • If C + m C − m <
0, there are N =min {| C + m | , | C − m |} Dirac cones and || C + m | −| C − m || chiral surface modes. • If C + m C − m ≥
0, there are | C + m | + | C − m | chiralsurface modes.Note that the chiral surface modes are protected bynot only mirror but also the (total) Chern numberof the system. In fact, even if the mirror symmetryis broken, the chiral modes are still preserved dueto nonzero Chern number.Lastly, we remark that there exist hinge modeswhen a mirror-preserving hinge is made.5. Unitary glide { M | a i } . Its surface state has beendiscussed in Ref.[48, 79, 80], and more recently inRef.[81, 82], which has an odd number of Diraccones on the glide-symmetric plane. This surfacestate can also be understood from the minimal dec-oration of { M | a i } , which resembles that of theanti-unitary translation. Thus they should sharethe same type of surface states. These Dirac conescan appear at the glide-invariant lines in the sur-face BZ, and an even number of Dirac cones canannihilated with each other.We remark that there exist chiral hinge modeswhen a glide-preserving hinge is made.6. Anti-unitary glide { M | a i } · T . As ( { M | a i } · T ) = { E | a i } , the weak invariant δ w,i = 2 Z when the invariant δ ( { M | a i } · T ) = Z , whichmeans there are 2 Z chiral surface modes on a glide-symmetric plane.1 a i Ta i
1× 1× 1× |C m |× × t / (10) unitary screw {C n |t }, n=2,3 , n T, n=2,4,6(8) inversion (9) S (7) anti-unitary screw {C n |t}T, n=2,4,6(5) unitary glide (6) anti-unitary glide inversion/S center: rotation/screw axis: mirror/glide plane: t / chiral hinge mode:2D surface mode: Z× FIG. 10. Surface states in MSGs, with the corresponding symmetries marked above the plots. The 2D surface modes can beeither sloped-like chiral surface modes in (1),(6),(10) or Dirac cones in (2),(3),(4),(5).
7. Anti-unitary screw { C n | t } · T, n = 2 , ,
6. On atypical cylinder termination, there are 2, 4, and 6 C n -symmetric chiral edge modes on the side sur-face, respectively. Note the top surface breaks the { C n | t }· T symmetry and thus has no gapless modesin general.8. Inversion. There is one inversion-symmetric chiralhinge mode on a surface termination that preservesthe inversion symmetry. States with δ ( P ) = 1 havenon-trivial axion angle θ = π and are usually re-ferred to as “axion insulators”. The axion angle θ is quantized to 0 or π when there exist anti-unitaryproper symmetries (including TRS) or unitary im-proper symmetries, and systems with θ = π can beseen as axion insulators in a generalized sense.9. S . There is one S -symmetric chiral hinge modeon a surface termination that preserves the S sym-metry.10. Unitary screw { C n | t } , n = 2 , , ,
6. In the sur-face states shown in Fig.10(10), we set t = n a i for simplicity, and there are n chiral surface modesrelated by { C n | n a i } on the side surface when δ ( { C n | n a i } ) = 1.Generally, the screw vector could be t = ml a i , where m, l are coprime integers. On the side sur-face, there are δ w,i chiral surface modes related by { C n | ml a i } per lattice vector, which can be seenfrom the invariant. As both δ ( { C n | ml a i } ) and δ w,i are integers and { C n | ml a i } l = { ( C n ) l | m a i } , δ w,i should be a multiply of l and δ ( { C n | ml a i } ) = ml δ w,i . Thus { C n | t } shares the same surface statewith the unitary translation, similar to the anti-unitary glide.2 Appendix D: Decorations, second encounter
In this section, we classify the TCs into three typesand discuss them in terms of their invariants. As canbe seen from the general form of TCI classifications, i.e., Z m × Z l or Z l × Z , there are three types of decorationsof TCs:1. Mirror Decorations. Mirror Chern insulators with( C + m , C − m ) = ( n, − n ), i.e., zero net Chern numbers,decorated on mirror planes, while all non-mirror 2-cells are not decorated, with an example shown inFig.11(a).2. (Translation) Z decorations. Decorations withnonzero weak (translation) invariants, which meansthe Chern number per unit cell is nonzero, as shownin Fig.11(b).3. Z decorations. Decorations that are not protectedby (unitary) integer translation symmetries (butsome are protected by half magnetic translations).They have zero net Chern number per unit cell, i.e.,zero weak invariants, and nontrivial Z invariants.As discussed in the main text, these Z decorationsare axion insulators, characterized by the π axionangle. We plot a Z decoration of MSG P mm Z decorations can also be further divided into twotypes depending on whether they have mirror sym-metries. When there are no mirror planes, the Z decorations become trivial after doubling, whilewhen there are mirror planes, the doubling of a Z decoration followed by bubble equivalence be-comes a mirror decoration, where all mirror planesare decorated with ( C + m , C − m ) = ± (1 , − Z decoration for simplicity,which is an abuse of notation, as the decoration ac-tually does not become trivial but transforms intoanother type of decoration after doubling.When mirror planes are present, this Z decora-tion must be chosen as a generator of classifica-tions, though its original contribution to the to-tal classification, i.e., the Z factor, is absorbedinto Z m in a way that 2 ¯ Z ∼ (cid:80) mi =1 ¯ M i , where¯ M i ’s are generators of mirror decorations and ∼ means up to bubble equivalence. More specifi-cally, if there are m mirror planes, the m gener-ators for the classification Z m should be chosen as { ¯ Z , ¯ M , ¯ M , · · · , ¯ M m − } .The three types of decorations can be well character-ized by their invariants:1. Mirror Decorations. As discussed above, this typeof decorations are mirror Chern insulators with C + m = − C − m on all mirror planes. Since the netChern number is zero for all 2-cells, all invariantsexcept mirror Chern numbers are zero. C m =(1,1) Mirror planeC m =(1,1) C=1C=1 (a) mirror decoration (b) translation decoration (c) Z decoration FIG. 11. Three types of decorations. (a). Mirror decoration.(b). Z or translation decorations. (c). Z decorations. This type of decorations can be defined as thosewith zero net Chern number on every 2-cell, whichis enough to distinguish them from the other twotypes of decorations.2. (Translation) Z decorations. As mentioned be-fore, this type of decorations are protected by(unitary) integer translation symmetries, thus theymust have nonzero weak invariants. Besides, otherinvariants, including the mirror Chern numbers and Z invariants, can also be nonzero.When mirror planes are present, Z decorations canhave mirror Chern insulators with C + m (cid:54) = − C − m onmirror planes, which have nonzero net Chern num-ber and do not belong to the mirror decorations.Moreover, when the center of a symmetry operationwith Z invariant, e.g., S , is passed by a decoratedlayer with odd Chern number, the Z invariant isnonzero for this Z decoration. In fact, a Z decora-tion just becomes another Z decoration when addedwith a mirror or Z decoration.In a word, Z or translation decorations are thosewith nonzero net Chern number per unit cell.3. Z decorations. Contrary to the Z decorations, thistype of decorations must have zero weak invariants.Note the “weak” invariants refer to those protectedby unitary integer translations, but not magnetichalf translations in type-4 MSGs.For this type of decorations, all Z -type invariants(except mirror Chern numbers), including weak in-variants together with unitary screw invariants andanti-unitary glide invariants, are all zero, while all Z invariants must be nonzero, i.e. δ ( g ) Z = 1.This can be seen with the help of the homo-morphism between symmetries and invariants, i.e.,Eq.(B1). Firstly, a Z decoration must have at leastone nonzero Z invariant of some symmetry g , i.e., δ ( g ) = 1. Then suppose there is another symmetry g having Z invariant. Their product, denoted as g = g · g , must have Z or trivial invariant. This isbecause g and g , which have Z invariant, must beunitary improper or anti-unitary proper, and theirproduct g must be unitary proper or anti-unitaryimproper, and these two symmetry operations have3 Z or trivial invariants, as shown in Table.III. If δ ( g ) = 0, then δ ( g ) = δ ( g ) + δ ( g ) = 1 + 0 = 1,which is forbidden in Z decorations no matter g has Z or trivial invariant. Thus we must have δ ( g ) = 1. Similarly, for any other g with Z in-variant, we have δ ( g ) = 1.As a result, the values of all Z invariants arebonded and equal to 1 for the Z decoration, whichcan be called as the “axion invariant” and takethe form of 3D magnetoelectric polarization P [49].This dictates that for any MSG, there is eitherone independent Z decoration (when δ ( g ) Z = 1),i.e., an axion insulator, or no Z decoration (when δ ( g ) Z = 0). (1,0) (0,-1) (2,0) (0,-2) (1,-1) (1,-1) double bubbleequivalent (c)(a) (b)(d) (2,0) (0,-2) FIG. 12. The Z decoration becomes a mirror decoration withall mirror planes decorated when doubled and transformedinto an LC using the bubble equivalence. There is another interesting and significant prop-erty for the Z decorations as mentioned previ-ously. When there are mirror planes, the doublingof a Z decoration followed by the bubble equiv-alence becomes a mirror decoration with all mir-ror planes decorated. This can also be understoodfrom the invariants. As demonstrated before, the Z decoration has all Z invariants nonzero, andmirror symmetries can have the net Chern numberas their corresponding Z C-invariants. This indi-cates that for a Z decoration, each mirror 2-cellis decorated with C total = C + m + C − m = ± C + m , C − m ) = ( ± ,
0) or (0 , ± C + m , C − m ) = ( ± ,
0) or (0 , ± C + m , C − m ) by ± (1 , C + m , C − m ) = ± (1 , −
1) on each mirror 2-cell, whichis a mirror decoration with all mirror planes deco-rated.To summarise, Z decorations are those with zeronet Chern number per unit cell but nonzero netChern number on some 2-cells.4 Appendix E: Layer and non-layer constructions
Recall that an overwhelming majority of TCs in non-magnetic SGs are layer constructions(LCs), with excep-tions in only 12 SGs, which are called non-layer construc-tions (non-LCs)[30, 32]. Those non-LCs built by 2D TIshave 2D planes that are only partly decorated, namely,the 2-cells decorated with 2D TIs do not extend to the en-tire 2D plane. On the contrary, there is a vast number ofnon-LCs in (type-1,3,4) MSGs. It is difficult and also notnecessary to exhaustively find all non-LCs in all MSGs,and here we give three representatives for demonstration.We remark an interesting property of non-LCs inMSGs. Unlike the non-LCs in non-magnetic SGs whichhave incomplete 2D decorated planes, here in MSGsall the non-LCs we know have complete 2D decoratedplanes, which split into small 2D (mirror) Chern insula-tor pieces with different signs of (mirror) Chern number. (1) Pmm2 C T xyT·t (2) P2’/m (3) P C C C FIG. 13. Three examples of non-layer constructions(non-LCs). (1). The non-LC in
P mm
2, which has C z , M x , and M y . (2). The non-LC in P (cid:48) /m , which has C · T . (3). Thenon-LC in P C
4, which has C z and anti-unitary half trans-lation T · t , where t = { E |
12 12 } . Note in (1) and (2),the side surfaces of the unit cell are also decorated, which weomit for simplicity. • The first representative non-LC is the Z decora-tion of type-1 MSG P mm
2, which has C z and twoorthogonal mirrors M x and M y anti-commutingwith each other.Two 2-cells related by C symmetry on the samemirror plane have the same directional chiral edgemodes on the 1-cell they intersect, i.e., the twored/green curves in Fig.13(1). Thus decorat-ing only one mirror plane with C-building blocksbreaks the no-open-edge condition, and the othermirror plane must also be decorated to compensatethe edge modes on the C -axis. These decorated2-cells are fixed on mirror planes and cannot bemoved together to trivialize. This decoration couldbe seen as the simplest one among all non-LCs inMSGs, where only planes in the x and y directionsare decorated non-uniformly, with adjacent pieceson a 2D plane having opposite Chern numbers. • The second representative non-LC is the Z deco- ration of type-3 MSG P (cid:48) /m ( P (cid:48) /m and P (cid:48) /m have similar non-LCs), where the crucial symme-try operation that makes it a non-LC is the anti-unitary rotation C y · T . The mirror planes definedby M y is split by C y · T into two 2-cells, where thetwo edge modes related by C y · T have the samedirection and can not cancel with each other on the1-cell they meet. This forces the 2D plane verticalto mirror plane to be decorated as well, in order tosatisfy the no-open-edge condition. • The third representative non-LC is the Z decora-tion of type-4 MSG 3.6 P C
2. In this case there isno mirror symmetry, but the magnetic translation T · { E |
12 12 } in the diagonal direction makes thedecoration a non-LC.5 Appendix F: A brief review of the symmetry-basedindicator theory
Symmetry-based indicator (SI) theory uses the sym-metry data at high-symmetry-points (HSPs) in the BZto diagnose the band topology of a system. The sym-metry data composes of the irreducible representations(irreps) of the valence bands at HSPs in SGs, or irre-ducible co-representations (coirreps) in MSGs, and thecollection of coirreps at all HSPs is denoted as the bandrepresentation (BR). SI theory compares the differencebetween two linear spaces, i.e., the space of all atomicinsulators (AIs) and the space of BRs that satisfy thecompatibility relations between all HSPs. A BR havingzero SI can be decomposed into an integer combinationof AIs, while a BR with nonzero SI cannot.AIs are constructed by placing non-interacting atomsin real space, which have flat dispersions. For a spe-cific MSG, the BR of its AIs form a finitely generatedAbelian group Z l , denoted as { AI } , where l is the num-ber of generators of this group. We can construct { AI } byfirst choosing specific Wyckoff positions in the real spaceand then placing orbits on them, where the orbits havesymmetries of the site symmetry group of the Wyckoffpositions and can be represented by a coirreps of the sitesymmetry group, with details shown in Appendix.M.Similarly, the linear space { BS } composes of the BRsthat satisfy the compatibility relations between all HSPs,which ensures no (symmetry-protected) band crossingsat high-symmetry lines or planes connecting HSPs, al-though gapless points could exist at generic momenta.For a given MSG, these two Abelian groups have thesame dimension[86], and the SI group is defined as thequotient group X BS = { BS } / { AI } = Z n × Z n × . . . Z n l ,which is a finite Abelian group.Practically, { BS } can be derived from { AI } and doesnot need to be calculated explicitly. For a given MSG,arrange its { AI } bases { a , a , . . . , a n } into a matrix A and do the Smith normal form decomposition: A n × m = a a ... a n = L n × n M n × m R m × m = L × d d . . .0 d l ... 0 0 ... b b ... b n ... b m ⇒ L − a a ... a n = d b d b ... d n b n (F1) where the nonzero diagonal terms { d , d , . . . , d l } of M are arranged in ascending order and d i | d i +1 , with theterms greater than 1 corresponding to the SI group Z d k × Z d k +1 × · · · Z d l . Each row b i of the right matrix R is abasis of { BS } , and a (cid:48) i = d i b i gives the independent basisof { AI } . The rows in R that correspond to the SI group,i.e., b k , b k +1 , . . . , b l , are called nontrivial BSs, and musthave nontrivial indicators values which generate the SIgroup.In the next section, we derive the formulas of SIs inMSGs using physical bases such that their values corre-spond to some topological invariants. However, we re-mark here that the formulas of SIs do not have a uniquechoice, and Smith form decomposition naturally gives aset of SI formula, i.e., the i -th column of R − gives riseto the corresponding z i indicator, as shown below.Given an arbitrary band representation B , first decom-pose it using the { AI } bases: B = c · AI = ˜ c · M · R, (˜ c = cL ) (F2)The indicators are then given by z i = ( B · R − ) i mod n i (F3)which means the i -th column of R − gives the z i indi-cator, because z i corresponds to the numerator of thefractional number in the decomposition of B .6 Appendix G: Generating SIs in MSGs
In Table.IV, we list the generating SIs in 1651 MSGs,and their explicit formulas and correspondence to TCIclassifications as well as topological invariants are de-rived in the following subsections. The word “generat-ing” we adopt here means the SIs in all other MSGs canbe generated using the combinations of these SIs, withonly 10 exceptions denoted as “corner cases”, which willbe introduced in Appendix.H. We also include the type-2 generating MSG P (cid:48) here for completeness, which isomitted in the main text where only type-1,3,4 MSGs areconsidered. MSG MSG type X BS SI P Z , , , z P, , z P, , z P, , z P P n, n = 2 , , , Z n z nC P n/m, n = 2 , , , Z n,n,n z + nm, , z − nm, , z + nm,π P Z , , z ,S , z , Weyl , z C P mmm Z , , , z (cid:48) P, , z (cid:48) P, , z (cid:48) P, , z (cid:48) P P /mmm Z , , z (cid:48) P, , z +4 m,π , z P /mmm Z , z +6 m,π , z P (cid:48) Z z (cid:48) ,S P nc (cid:48) c (cid:48) , n = 2 , , Z n z (cid:48) nC TABLE IV. Generating SIs and their corresponding MSGs.We use a simplified notation to represent the SI group, i.e., Z n ,n ,... = Z n × Z n × . . . .
1. Generating SIs in type-1 MSG
In Ref.[55], Ono and Watanabe derived SIs in a fewtype-1 key MSGs. Here we do a more detailed investi-gation, including not only the SI formulas but also theircorrespondence to topological crystals. a. MSG 2.4 P , X BS = Z , , , , Classification= Z × Z MSG 2.4 P P , and its SIscan be defined using parities at 8 time-reversal invariantmomenta (TRIMs): z P, = (cid:88) k ∈ TRIM ,k = π
12 ( N − k − N + k ) mod 2 z P, = (cid:88) k ∈ TRIM ,k = π
12 ( N − k − N + k ) mod 2 z P, = (cid:88) k ∈ TRIM ,k = π
12 ( N − k − N + k ) mod 2 z P = (cid:88) k ∈ TRIM
12 ( N − k − N + k ) mod 4 (G1)where N ± k is the number of valence bands having positive(negative) parity. The first three z P,i indicators diagnose the Chernnumber mod 2 on k i = 0 , π planes, while the z P indica-tor can be seen as the sum of Chern number on k z = 0 , π planes.When Z = 1 ,
3, the Chern number on k z = 0 and k z = π plane must differ by an odd number, which indicatesan odd number of Weyl points exist between k z = 0 and k z = π . An equal number of Weyl points of oppositechirality exist between k z = 0 , − π planes, enforced bythe inversion symmetry.The 4 nontrivial BSs that corresponds to Z , , , are b = V − V + X − X + Y − Y − Z + Z b = Y − Y − Z + Z b = X − X − Z + Z b = Z − Z (G2)Their indicators are calculated to be (0012), (0110),(1010), (0013), which indeed generate the group Z , , , .The minus sign in the BSs may be confusing, and onecan add proper AIs to them to make the coefficients allpositive, which does not change SIs as AIs have zero SIs.The TCI classification of P ¯1 is Z × Z , where thethree Z indexes correspond to translation decorations,while the Z index is the Z decoration protected byinversion. The generators of these decorations haveweak and inversion invariants ( δ w, , δ w, , δ w, , δ ( P )) =(100 , , (010 , , (001 , , (000 , z P,i = δ w,i mod 2 , z P = 2 δ ( P ) (G3)which means z P = 2 states have δ ( P ) = 1, i.e., they have1D chiral hinge modes on inversion-preserving surfaces.Note that z P = 1 , b. P n, X BS = Z n , Classification= Z P n, n = 2 , , , C n rotations as generators. Be-cause the C n eigenvalues must be the same along therotation axis, the Chern number on k i = 0 and k i = π plane must be the same, and we take the Chern numberon k i = 0 plane as the Z n indicator, where i denotes therotation direction. The classification group Z is gener-ated by a LC with C = 1 layers on x i = n planes, whichhas weak invariant δ w,i = 1.The formulas for calculating Chern numbers (mod n)7 SI=(0,0,1,2) SI=(0,0,0,1)SI=(0,0,0,2) x yz x yzC=+1 C=-1C=+1 k k k SI=(0,0,1,0) x yz C=-1 SI=(0,1,0,2) x yz
SI=(1,0,0,2) x yz - ++ ++++ - k k k - ++ ++ - k k k -- -- ++-+ ++++ +- k k k +-+-- ++++ k k k -- ++ + k k k - - ++ ++ k k k +++++ +++ (c)(a) (b)(f)(d)(e) C=+1 C=+1C=+1C=+1C=+1C=-1 C=-1C=+1
FIG. 14. (a)-(e). Layer constructions and their compatiblesets of irreps as well as the corresponding SIs in P
1. (f). TheWeyl state and a compatible set of irreps in P using C n eigenvalues are derived in Ref.[49]: C : ( − C = (cid:89) l ∈ occ . B lC (Γ) B lC ( X ) B lC ( Y ) B lC ( M ) C : e iπC/ = (cid:89) l ∈ occ . ( − B lC (Γ) B lC ( M ) B lC ( Y ) C : e i πC/ = (cid:89) l ∈ occ . ( − B lC (Γ) B lC ( K ) B lC ( K (cid:48) ) C : e iπC/ = (cid:89) l ∈ occ . ( − B lC (Γ) B lC ( K ) B lC ( M ) (G4)where B lC n ( k ) represent the C n eigenvalue of the l -thband at C n -invariant point k , and the HSPs used in theseformulas are plotted in Figure 15. We can take logarithmto each side of the equation to turn them into the morefamiliar summation and mod n form, and denote themas z nC : z C = (cid:88) l ∈ occ ln( B lC (Γ) B lC ( X ) B lC ( Y ) B lC ( M )) / ( iπ ) mod 2= (cid:88) k ∈ Γ ,M,X,Y
12 ( N − k − N + k ) mod 2 z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( M ) B lC ( Y )) / ( i π z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( K ) B lC ( K (cid:48) )) / ( i π z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( K ) B lC ( M )) / ( i π z nC indicators correspond to the weak invariant: z nC = δ w,i mod n (G6) ГY X ГM Y XMГ K K ’ K ’’ Г KMM ’ (a) C (b) C (c) C (d) C FIG. 15. HSPs used to calculate Chern number mod n on 2DBZ In P P
3, as S = C − and S = C − on k z =0 , π planes, they can also be used to calculate the Chernnumber. The modified formulas are S : z C = (cid:88) l ∈ occ ln( − B lS − (Γ) B lS − ( M ) B lC ( Y )) / ( i π S : z C = (cid:88) l ∈ occ ln( B lS − (Γ) B lC ( K ) B lP ( M )) / ( i π z nC indicators us-ing coirreps of double MSGs, we need to choose theproper representation matrices s.t. the SU(2) part of C n rotations are aligned, i.e., U ( C ) = U ( C ) , U ( C − ) = − U ( C ) ,U ( C ) = U ( C ) , U ( C − ) = U ( C − ) ,U ( C ) = U ( C ) , U ( C − ) = − U ( C ) (G8)where U denotes the SU(2) matrix. For S , S , theserelations become U ( S − ) = U ( C ) , U ( S − ) = U ( C ) , U ( S − ) = U ( P )(G9)Note for S , there is no − z C formula, becausewe are using P instead of C , which satisfies P − = P instead of C − = − C when SU(2) matrices are consid-ered. c. P n/m, X BS = Z n,n,n , Classification= Z P n/m represents four MSGs: P /m , P /m , P P /m ), and P /m . All these MSGs have mirror planes8vertical to the C n rotation axis. Assume the rotation isalong x i direction. There are two mirror planes in theBZ, i.e., k i = 0 , π , which allows us to define four Chernnumbers: C ± m, and C ± m,π , where ± means the Chernnumber defined on the mirror ± i sector, respectively. Itcan be proven that only three of them are independent,which correspond to the three Z indexes in the TCI clas-sification.The indicators can be chosen as any three independentmirror Chern numbers mod n, and here we take the firstthree: z + nm, , z − nm, , z + nm,π . The mapping between invari-ants and indicators are simple: z ± nm, /π = C ± m, /π mod n (G10) a. Example: P /m We take P /m as an example,which has three nontrivial BSs as: b = − Γ + Γ − M + M + R − R − X + X + 2 Z + 3 Z − Z − Z + Z b =Γ − Γ − M + M − R + R + X − X − Z − Z + 2 Z + 2 Z − Z b = Z + Z − Z − Z (G11)Their SIs are calculated to be { z +4 m, , z − m, , z +4 m,π } =(3 , , , (0 , , , (0 , , Z , , group. b. Example: P /m , X BS = Z , P /m has ascrew symmetry { C z | , , / } and SI group Z , . ItsTCI classification is Z , whose generators can be takenas a translation decoration (placing ( C + m , C − m ) = (1 , z = 1 / , / Z decoration, whichis a non-LC similar to the non-LC in P mmm , as shownlater.The SIs can be taken as C +4 m, , C +2 m,π . The Chern num-ber on k z = π plane can only be calculated by C z , butnot C z -screw, as C z -screw does not commute with M z .The nontrivial BSs are b = 2 M + 2 M − M − M + R − R + X − X b = M + M − M − M (G12)which has SI=(1,0) and (0,1).Similar argument hold for P /m and P /m . P /m has indicator group Z , and can be taken as C +6 m, , C +3 m,π , while P /m has indicator group Z , andcan be taken as z (cid:48) P . d. MSG 81.33 P , X BS = Z , , , Classification= Z × Z P S symmetry. Its TCI classification is Z × Z ,which correspond to a translation and a S decoration,respectively. The translation decoration is placing 2Dlayers with Chern number C on z = n + 1 / S decoration is composed of 2D layers with Chern number 1 on z = n planes and layers with Chernnumber -1 on z = n + planes.The SI group of P Z , , . Because S symmetry isequivalent to C − on k z = 0 , π planes in the BZ, S helpsdiagnose the Chern number mod 4 on these two planes.As a result, the Z index of translation decoration turnsinto the Z factor in the SI group.The first Z in the SI group corresponds to Weyl states,which exist when the Chern number on k z = 0 , π planesdiffer by 4 n +2, indicating an even number of Weyl pointsbetween k z = 0 , π planes. Note S = C , which requiresthe Chern number on k z = 0 , π planes to be equal mod2, and cannot differ by 4 n + 1 or 4 n + 3.The second Z corresponds to the S decoration, andcan be calculated by the formula: z ,S = 12 (Re µ − Im µ ) mod 2 µ = 1 √ (cid:88) i ∈ occ. (cid:88) k ∈ K S β i ( k ) (G13)where β i ( k ) = e α π i ( α = 1 , , ,
7) are the S eigenvaluesat four S -invariant momenta K S . It can be proven that µ = ± ± i, µ = ± , ± i for the S decoration. The z ,S indicatormaps the blue dashed lines in Fig.16 to z = 1 and theblack dashed line to z = 0. This definition is stableagainst adding AIs to the BS.This z ,S also maps µ = ± (1 − i ) to z ,S = 1, while µ = ± (1 + i ) to z ,S = 0. These µ values can only betaken in Weyl states, which means that the z indicatorclassifies Weyl states into two classes. We can define anew z , Weyl indicator to represents these Weyl states: z , Weyl = Re( µ ) mod 2 (G14)
21 Rei2iIm
FIG. 16. .Possible values of µ . Orange dots correspond toAIs or the translation decorations, red dots correspond to the S decorations, and black dots correspond to Weyl states. a. Meaning of the z ,S and z , Weyl indicator.
Fromthe TCI classification Z × Z , we can naturally extract Z × Z as indicators, where the Z indicator is used toidentify the S decoration.For the translation decoration, 2D layers with Chernnumber C=1 are placed on z = n +1 / S eigenvalues on k z = 0 , π planes are opposite, whichimplies that µ = 0. For the S decoration, 2D layerswith Chern number C=1 placed on z = n planes havethe same S eigenvalues on k z = 0 , π planes, while the2D layers placed on z = n + planes have opposite S eigenvalues on k z = 0 , π planes. As a result, only the z = n layers contribute to µ .We tabulate the possible choices of S and C eigen-values for a 2D layer with C = 0 , , , momenta Γ M X sum of S eigenvalues / √ S − S − C eigenvalue e iπ/ e i π/ i / − i i -1 / 1 e i π/ e iπ/ i / − i ie − iπ/ e − i π/ i / − i − ie − i π/ e − iπ/ i / − i − ie iπ/ e − iπ/ − i / i e − iπ/ e iπ/ − i / i e i π/ e − i π/ − i / i − e − i π/ e i π/ − i / i − e iπ/ e iπ/ i / − i i e − i π/ e − i π/ i / − i − − ie i π/ e − iπ/ i / − i e − iπ/ e i π/ i / − i e i π/ e i π/ − i / i − ie − iπ/ e − iπ/ − i / i − ie iπ/ e − i π/ − i / i e − i π/ e iπ/ − i / i S and C eigenvalues for a2D layer with Chern number C = 0 , , , z C = (cid:80) l ln( − B lS − (Γ) B lS − ( M )( − B lC ( Y )) / ( i π ) mod4, where the minus sign in front of B C comes from theSU(2) matrix alignment for C , in order to be consistentwith the Bilbao irrep convention.From Table.V we can see that µ = ± , ± i for the S decoration, while µ = 0 , ± ± i for AIs. For Weylstates, the Chern number on k z = 0 /π planes differ by 2.We can enumerate the possible choices of S eigenvaluesthat are compatible with the Weyl states. Note that the C eigenvalues on k z = 0 /π planes need to be the same,with S − = e iπ/ , e − i π/ corresponding to C = i and S − = e − iπ/ , e i π/ corresponding to C = − i . We findthat all Weyl states has µ = ± ± i . As a result, z ,S and z , Weyl are well-defined and can successfully identifythe S decoration and Weyl states. b. Nontrivial BSs The three nontrivial BSs corre-spond to Z , , are: b = Z − Z b = Z − Z b = Γ − Γ + Z − Z (G15)Their indicators ( z , Weyl , z ,S , z ) are calculated to be(1,1,0), (1,0,0), and (0,1,1). c. Type-2 MSG P (cid:48) in type-2 MSG has indicator Z defined as z (cid:48) ,S = − µ mod 2, which representsSTIs, and can be induced from the Weyl states in type-1MSG. This is because a time-reversal partner of a certainstate in type-1 MSG has complex-conjugated µ value.As a result, only µ = ± (1 ± i ) states, which correspondto Weyl states, can have z (cid:48) ,S = 1, which corresponds tothe STI. d. MSG 82.39 I , X BS = Z , , , Classification= Z × Z . The indicator of I P
4, and can betaken as z , Weyl , z ,S , z = z C /
2. Because I ¯4 is body-centered, the Chern number on k z = 0 plane can onlytake even number, which enforces z C to become a Z indicator.The z , Weyl needs further clarification. Because theTCI classification generators, i.e., translation and S dec-oration, correspond to the z C / z ,S separately,the remaining Z indicator must correspond to a gap-less state. On the other hand, the z , Weyl indicator to-gether with z ,S , z C / Z , , group, we conclude that the z , Weyl indicator is applica-ble. z , Weyl = 1 corresponds to a Weyl semimetal withWeyl points lying S -symmetrically around Γ at genericmomenta, and two Weyl points connected by S havingopposite chirality. e. MSG 47.249 P mmm, X BS = Z , , , , Classification= Z P mmm = { , C x , C y , C z , P, M x , M y , M z } .Because there are two mirror planes in each direction,i.e., x, y, z = 0 , /
2, the TCI classification of
P mmm is Z , with each mirror plane corresponding to one Z .Notice three C forbid the translation decoration, i.e.,a pure stacked C=1 state, on each direction, but onlymirror decorations are allowed, i.e., ( C + m , C − m ) = ( n, − n )states on x, y, z = n, n +1 / Z canbe taken as five mirror decorations, plus a Z generator.The SI group of P mmm is Z , , , . Because M x , M y ,and M z anti-commute with each other, irreps in P mmm are all two-dimensional, with the same inversion eigen-values and opposite mirror eigenvalues, forming effectiveKramer pairs. As a result, we can modify the indicatorformula in P N ± k to the number of Kramer0pairs: z (cid:48) P,i = (cid:88) k ∈ TRIM ,k i = π
14 ( N − k − N + k ) mod 2 z (cid:48) P = (cid:88) k ∈ TRIM
14 ( N − k − N + k ) mod 4 (G16)The four nontrivial BSs correspond to Z , , , are: b = U − U + X − X − Y + Y + Z − Z b = Y − Y − Z + Z b = X − X − Z + Z b = Z − Z (G17)Their indicators are calculated to be (0,1,0,2), (0,1,1,0),(1,0,1,0) and (0,0,1,1). P mmm X BS = Z , , , Classification= Z M deco Z weak 2 i Cm Cm Cm Z M (100; 0) 1002 (000) 0 0 0 0 1¯11¯1 0000 0000 M (100; ) 1000 (000) 0 0 0 0 1¯1¯11 0000 0000 M (010; 0) 0102 (000) 0 0 0 0 0000 1¯11¯1 0000 M (010; ) 0100 (000) 0 0 0 0 0000 1¯1¯11 0000 M (001; 0) 0012 (000) 0 0 0 0 0000 0000 1¯11¯1 M (001; ) 0010 (000) 0 0 0 0 0000 0000 1¯1¯11TABLE VI. Topological invariants and SIs of P mmm decora-tion generators. M ( nml ; d ) denotes an LC with Miller indices( mnl ) and the distance d to the origin point. Note we list all6 mirror decorations for completeness, and any 5 of them plusthe Z decoration can be chosen as the classification genera-tors. a. Interpretation of the SI We list the topologicalinvariants of decoration generators in Table.VI. The 6LCs are realized by placing ( C + m , C − m ) = (1 , −
1) stateson x, y, z = n, planes, respectively. Their SIs can becalculated by assigning compatible irreps similar to P ¯1.The correspondence between SIs and invariants is z (cid:48) P,i = C + m,k i = π mod 2 , z (cid:48) P mod 2 = δ ( P ) (G18) z (cid:48) P = 1 , δ ( P ) = 1, while z (cid:48) P = 2corresponds to states with helical hinge modes, similar tothe case of type-2 MSG P ¯11 (cid:48) which has the same Z , , , SIs.The Z decoration is a non-LC patched using 2D mir-ror Chern insulators pieces, as discussed in the maintext. Although it is not straightforward to get the SIof this decoration, we can understand it by first dou-bling it and them using bubble equivalence to trans-form it into an LC, which has mirror Chern number(1 , −
1) on x, y, z = n, n + 1 / k x , k y , k z = 0 planes is (2,-2), while on k x , k y , k z = π plane is (0,0), which has SI=(0,0,0,2).As bubbles has trivial SI, we conclude that the Z dec-oration has SI=(0,0,0,1) or (0,0,0,3). In Appendix.L, wecalculated the 3D BHZ model in detail, whose symmetryand mirror Chern numbers are compatible with the Z decoration when M ∈ (1 , Z decoration must have the same paritiesat all TRIMs and thus the same z (cid:48) P indicator as 3D BHZmodel, which means z (cid:48) P = 3 (G19) f. MSG 123.339 P /mmm, X BS = Z , , ,Classification= Z P /mmm = { , C x , C y , C z , C z , C , , C , , P,M x , M y , M z , M , M , ... } . Its TCI classification is Z ,which can be interpreted as four mirror decorations alongx/y and z directions ( C z makes x and y equivalent), eachhaving two mirror planes at 0 and , and one mirror dec-oration along (110)/(1¯10) direction ( C z also makes (110)and (1¯10) equivalent).The Z indicator corresponds to the weak mirror Cherninsulator along x and y direction and can be taken as the z (cid:48) P, , i.e., the mirror Chern number mod 2 on k x = π plane.The Z indicator corresponds to the weak mirror Cherninsulator along the z direction and can be taken as themirror Chern number on the k z = π plane mod 4,i.e., z +4 m,π , calculated by the C eigenvalues.The Z indicator can be taken as z = 2 z (cid:48) ,S − z (cid:48) P mod 8 z (cid:48) P = (cid:88) K ∈ TRIM
14 ( N − k − N + k ) z (cid:48) ,S = − √ (cid:88) i ∈ occ. (cid:88) k ∈ K β i ( k )= (cid:88) k ∈ K S
12 ( n −√ k − n √ k ) (G20)where N ± k are the number of bands with parity ± z (cid:48) ,S = − µ defined by S symmetry, and n ±√ k isthe number of Kramer pairs at S -invariant TRIMs withtr [ D ( S )] = ±√
2. Note that z (cid:48) P and z (cid:48) ,S in the z formula should not mod 4 and mod 2, and their originalvalues are used to calculate z .The three nontrivial BSs correspond to Z , , are: b = − M + 2 M − R + R b = − M + M − R + R + Z − Z b = Z − Z (G21)Their indicators are calculated to be (1,2,0), (0,3,6),(0,1,3).1 P /mmm X BS = Z , , Classification= Z M deco Z weak 4 i ¯4 Cm Cm Cm Z
003 (000) 0 0 1 0 1 1¯100 1¯100 1¯1M(100; 0) 104 (000) 0 0 0 0 0 1¯11¯1 0000 00M(100; ) 100 (000) 0 0 0 0 0 1¯1¯11 0000 00M(001; 0) 016 (000) 0 0 0 0 0 0000 1¯11¯1 00M(001; ) 030 (000) 0 0 0 0 0 0000 1¯1¯11 00M(1¯10; 0) 004 (000) 0 0 0 0 0 0000 0000 2¯2TABLE VII. Topological invariants and SIs of P /mmm decoration generators. SI=(0,1,6) x yzC m =(1,-1) SI=(0,1,0) x yz k k k (a) LC(001;0) (b) LC(001;1/2) SI=(1,0,4) x yz
SI=(1,0,0) x yz (c) LC(100;0) (d) LC(100;1/2) C m =(1,-1) Г X M Z R A k k k Г X M Z R A k k k Г Г X X M M Z Z R R A A k k k Г Г X X M M Z Z R R A A (e) Z (f) C m =(1,-1) C m =(1,-1) SI=(0,0,3)SI=(0,0,6)
FIG. 17. The decorations of P /mmm . (a)-(d) are the mirror decorations and compatible sets of irreps. (e) is the Z decorationand (f) is its doubled mirror decoration, with mirror planes all having mirror Chern number (1 , − a. Interpretation of the SI We list the invariantsand SIs of decoration generators in Table.VII, and plottheir real space constructions in Fig.17. We attach pos-sible irreps at HSPs that are compatible with the LCs,with which SIs can be readily calculated.Similar to
P mmm , the Z decoration is a non-LCpatched by 2D mirror Chern insulator pieces. Its dou-bled and then bubbled state is an LC with mirror Chernnumber (2 , −
2) on k x , k y , k z , k , k = 0 planes and(0 ,
0) on k x , k y , k z = π planes, which has SI=(0 , , z = 3 or 7 for the Z deco-ration. In Appendix.L, we calculate the 3D BHZ model,whose symmetry is compatible with this Z decoration,and the z indicator is calculated to be z = 3 (G22)The indicator of the Z decoration can also be un-derstood in the following way. First, it has inversioninvariant δ i = 1, thus has inversion indicator z (cid:48) P = 1or 3. On the other hand, it is a two-band system thathas S invariant 1, thus it can be seen as a superpo-sition of δ ( S ) = 1 and δ ( S ) = 0 state in SG81. InSG81, a gapped δ ( S ) = 0 state has µ = ± ± i, δ S = 1 state has µ = ± , ± i . By enforcing their sum to be real (because z (cid:48) ,S is always real), wehave z (cid:48) ,S = 1 mod 2. Combining these two conditions, z can only take odd values for the Z decoration. g. MSG 191.233 P /mmm , X BS = Z , ,Classification= Z P /mmm has TCI classification Z due to four setsof independent mirror planes, including two sets of ver-tical mirrors (in x and (110) direction) and two sets ofhorizontal mirrors (at z = 0 and z = ).Its SI group is Z , . The first Z can be calcu-lated from the mirror Chern insulator in the z direction z +6 m,k z = π .The choice of Z is not obvious. We can define z m,S = z +6 m,k z =0 + z +6 m,k z = π and z (cid:48) P , with z m,S = 1and z (cid:48) P = 1 both correspond to the Z decoration, thusthey are not independent and merge into one Z indi-cator. Using the same method from Ref.[30], we have z mod 6 = z m,S z mod 4 = z (cid:48) P (G23)2which is equivalent to z = { z m,S + 3 [( z m,S − z (cid:48) P ) mod 4] } mod 12 (G24)We remark that the z indicator is stable when z m,S ischanged by 6 and z (cid:48) P changed by 4, i.e., z m,S , z (cid:48) P canfirst take mod and then insert into the z formula.The two non-trivial BSs of Z , are: b = − H + 2 H − K + 2 K + L − L + M − M b = 4 H − H + 4 K − K − K − L + 2 L − M + M (G25)which have Z , indicators (1,2), (4,9). P /mmm X BS = Z , Classification= Z M deco Z , weak 6 i Cm Cm Cm ¯210(2) Z
07 (000) 0 0 1 0 1¯1 1¯100 1¯1 M (100; 0) 06 (000) 0 0 0 0 2¯2 0000 00 M (001; 0) 12 (000) 0 0 0 0 00 1¯11¯1 00 M (001; ) 50 (000) 0 0 0 0 00 1¯1¯11 00 M (¯210; 0) 06 (000) 0 0 0 0 00 0000 2¯2TABLE VIII. Topological invariants and SIs of P /mmm dec-oration generators. The generators of decorations are listed in Table.VIII.For the four mirror decorations, the z indicator can becalculated by first read the z m,S from the mirror Chernnumbers, and then determine the z (cid:48) P by reducing the LCto P mmm . For the Z decoration, z = 7 is calculatedfrom the BHZ model, which has z m,S = 1 , z (cid:48) P = 3.Type-4 MSG 191.242 has Z indicator group and canbe defined similarly. h. MSG 147.13 P , X BS = Z , , Classification= Z × Z MSG P Z indicator, whose formula isnot obvious. Its SI group can be rewritten equivalentlyas Z × Z = Z × Z × Z = Z × Z .As S is equivalent to C on k = 0 , π planes, we cantake the Z indicator as z C , which corresponds to thetranslation decoration. The Z indicator is taken as the z P from P ¯1, with z P = 2 corresponds to δ ( P ) = 1,and odd z P correspond to Weyl states, where the Chernnumber on k z = 0 , π planes differ by 3 mod 6. In themapping table of P
3, we set the SI group to be Z , , andthe SI can be directly read from the weak and inversioninvariants. P Z × Z , where Z corresponds to the translation decoration and Z is the Z decoration, as shown in Table.IX.Four type-3 MSGs 162.77, 163.83, 164.89 and 165.95with P ¯3 as subgroup also have similar SIs. P ¯3 X BS = Z , Classification= Z × Z deco Z , weak 3 iZ
02 (000) 0 1 Z
50 (00¯1) 0 0TABLE IX. Topological invariants and SIs of P
2. Generating MSG in type-2 MSGs • MSG 81.34 P (cid:48) . This type-2 MSG has the follow-ing Z indicator: z (cid:48) ,S = − µ S = (cid:88) k ∈ K S
12 ( n −√ k − n √ k ) (G26)where n ±√ k is the number of Kramer pairs at S -invariant TRIMs with tr [ D ( S )] = ±√
2. Asshown in Ref.[30], z (cid:48) ,S = 1 represents a STI.We remark that all other SIs in type-2 MSGs canbe induced from type-1 generating SIs, althoughtheir correspondence to topological invariants couldchange because the definition of invariants is differ-ent in type-2 MSGs.
3. Generating MSGs in type-3 MSGs a. P nc (cid:48) c (cid:48) , X BS = Z n , Classification= Z There are three generating type-3 MSGs and their SIscan be seen as generalizations of SI formulas of P , P P , because their coirreps are two-dimensional with the same C n eigenvalues on k z = π plane. • MSG 27.81
P c (cid:48) c (cid:48) z (cid:48) C = 12 z C mod 2 (G27) • MSG 103.199 P c (cid:48) c (cid:48) z (cid:48) C = 12 z C mod 4 (G28) • MSG 184.195 P c (cid:48) c (cid:48) . z (cid:48) C = 12 z C mod 6 (G29)where z nC on the right hand side has not takenmodulo.The TCI classifications of these three MSGs are all Z ,i.e., the translation decoration. This decoration has even3weak invariant δ w, and the Chern number can only takeeven numbers on k z = 0 /π planes. The three P nc (cid:48) c (cid:48) allhave { C nz , { M x · T | } , { M y · T | }} , and when thereis a layer with Chern number C = m on z = N plane,there will be another plane with the same Chern numberon the z = N + plane, which enforces the weak invariant δ w, = 2 m .For MSG 27.81, as C z { M x · T | } = −{ M x · T | } C z , where the minus sign comes from the SU(2)matrices, and the eigenvalue of C z is ± i , which con-tributes an extra minus when moved outside T , the twolayers connected by { M x · T | } must have the same C z eigenvalues, which validates the z (cid:48) C formula.For MSG 103.199, we have C z { M x · T | } = { M x · T | } C − z . Because the eigenvalues of C z and C − z areconjugated, the two layers connected by { M x · T | } must have the same C z eigenvalues, which validates the z (cid:48) C formula. Similar argument holds for MSG 184.195.As a result, we conclude that for these three MSGs, the z (cid:48) nC indicator is valid and equals 1 when the weak invari-ant δ w, = 2, i.e., the mapping between weak invariantand SI is z (cid:48) nC = 12 δ w, mod n (G30)We remark that when there are C n , n = 2 , , T ( G x · T, G y · T ) symmetries, we have( G x/y · T ) = − k z = π plane, and it can be proventhat the Berry phase of half of the loop chosen Fig.15 isquantized to 0 or π , which enforces the Chern number of k z = π plane to be even numbers, as shown for SG 27,103 and 184 in Ref. [85]. a. z (cid:48) nC in other type-3 MSGs The z (cid:48) nC , n = 2 , P nc (cid:48) c (cid:48) , or have a different Bravais lattice. • z (cid:48) C . MSG 39.199 Ab (cid:48) m (cid:48) Ib (cid:48) a (cid:48) P c (cid:48) c (cid:48)
2, while MSG 54.342
P c (cid:48) c (cid:48) a and 56.369 P c (cid:48) c (cid:48) n are parent MSGs of P c (cid:48) c (cid:48) • z (cid:48) C . MSG 108.237 I c (cid:48) m (cid:48) has a different latticesfrom P c (cid:48) c (cid:48) , while MSG 130.429 P /nc (cid:48) c (cid:48) is a par-ent MSGs of P c (cid:48) c (cid:48) .The z (cid:48) nC indicator naturally apply in parent MSGs,because we can always use a group’s subgroup to calcu-late its SI, while the case of different Bravais lattice needfurther clarification, which we leave to the next section.
4. Induce SIs in other MSGs from generatingMSGs
To induce the SIs in all MSGs, we first find the maxi-mum unitary subgroup with nontrivial SI group and thenuse the unitary subgroup’s SI formula to extract effectiveSI formulas by examining nontrivial BSs. Notice that the generating MSGs we considered hereare all in a primitive lattice. For an MSG M having thesame crystalline symmetries but a complicated lattice,the SI group of M is usually a subgroup of the corre-sponding primitive lattice MSG M . One can induce theSI formula of M from M by applying the SI formulas of M to the nontrivial BSs of M and extract the effectiveSI formulas. On the other hand, one can also extract theSI formula from the quantitative mappings between TCIclassification generators and SIs, as a real space construc-tion of M is also compatible with M , and they have thesame mapping between invariants and SIs. Caution thatthe HSPs of M and M are not the same, but the strongSIs like z (cid:48) P remains stable under a change of Bravaislattice, which can be seen from the BZ folding process. a. 8 special MSGs with z (cid:48) P indicator There are 8 type-3 MSGs that have z (cid:48) P as SI but theircoirreps are not all twofold degenerate to form effectiveKramer pairs: MSG 83.45 P (cid:48) /m , 87.77 I (cid:48) /m , 124.354 P (cid:48) /mc (cid:48) c , 124.355 P (cid:48) /mcc (cid:48) , 127.391 P (cid:48) /mbm (cid:48) , 128.402 P (cid:48) /mn (cid:48) c , 128.403 P (cid:48) /mnc (cid:48) , and 140.545 I (cid:48) /mcm (cid:48) .We observe that the z (cid:48) P formula is valid because (i) theytake odd values for nontrivial BSs s.t. they can generatethe SI group, and (ii) they equal to zero for all AIs. Notethat the z (cid:48) P formula in these MSGs is only effective andmay become invalid when the origin point is changed.We fix the coordinate system by adopting the conventionof Bilbao.For these 8 MSGs, we need to determine the value of z (cid:48) P for the Z decorations. Take MSG 83.45 P (cid:48) /m asan example. It can be seen that its Z decoration is thesame as the Z decoration in P mmm if we set the latticebases a = a . Their common invariants are thus thesame with C m,k z = (1 , − , , , δ i = 1. As a result, theymust share the same parities at all TRIMs and thus have z (cid:48) P = 3, as proved using the BHZ model.MSG 87.77 I (cid:48) /m has the same symmetry operationsbut a body-centered lattice with P (cid:48) /m . Because z (cid:48) P is a strong index and is stable against the BZ folding, I (cid:48) /m must have the same mapping between invariantsand z (cid:48) P .The other 6 MSGs all have P (cid:48) /m or I (cid:48) /m as a sub-group, and their Z decorations are compatible with the Z decoration in P (cid:48) /m , i.e., have the same common in-variants. As a result, they also take z (cid:48) P = 3. b. The BZ folding process a. Inversion-symmetric BZ First, let’s consider a1D BZ with inversion symmetry, as shown in Fig.18. TheHSPs are two inversion-invariant points k = 0 , π . Weplot a schematic energy band, which are symmetric at k = 0 , π . When we double the lattice vector, the recipro-4 Г MXAX A k k k E k E �/2 folding (a) (b) (c) FIG. 18. (a), (b). Band folding on the 1D BZ. (c). HSPs of 2D BZ. cal lattice vector is halved and becomes G = π , leadingto a folded BZ. The original TRIMs k = 0 , π becomesequivalent, and k = π/ k = π/ k = π . Assume the parityin the unfolded BZ is D ( P ) , D π ( P ), then the parity atthe new TRIMS in the folded BZ is D (cid:48) ( P ) = (cid:18) D ( P ) 00 D π ( P ) (cid:19) ,D (cid:48) π/ ( P ) = (cid:18) (cid:19) (cid:117) (cid:18) − (cid:19) (G31)where (cid:117) means two matrices are similar. As a result, thesummation (cid:80) k ( N − k − N + k ) is unchanged when the BZ isfolded.Next, let’s consider a 2D BZ in Fig.18(c), which can bethe k z = 0 plane of a base-centered or body-centered 3Dlattice. The original BZ has 4 TRIMs Γ , M, X, X . Af-ter folding, Γ , M are folded together, and X, X are alsofolded. There arise two new TRIMs A, A . Similar tothe 1D case, the representation matrices of inversion at A, A are off-diagonal and have opposite inversion eigen-values. As a result, the summation (cid:80) k ( N − k − N + k ) is alsounchanged when the 2D BZ is folded.From the above analysis, we conclude that the z P indicator is stable when we change the Bravais lattice,so as z (cid:48) P . b. C -symmetric BZ The z C indicator is also sta-ble when the body-centered lattice is changed to thesimple (principal) lattice. In a body-centered lat-tice BZ in Fig.18(c), the reciprocal lattice vectors are(1 , , , (1 , , , (0 , , k z = 0 plane, there aretwo C -invariant HSPs Γ(0 , ,
0) and M (1 , ,
0) and a C -invariant HSP X ( , , z C indicator.When changed to the simple lattice, the reciprocal lat-tice vectors become the usual (1 , , , (0 , , , (0 , , k z = 0 plane, M is folded to Γ, X becomes a C -invariant HSP, and there appears a new C -invariantHSP A ( , , D Γ ( C ) , D M ( C ) , D X ( C ) (G32) After folding to the simple lattice BZ, the three new HSPshave the following doubled representations: D (cid:48) Γ ( C ) = (cid:18) D Γ ( C ) 00 D M ( C ) (cid:19) , D (cid:48) X ( C ) = (cid:18) D X ( C ) 0 (cid:19) ,D (cid:48) A ( C ) = (cid:18) − (cid:19) (G33)In the z C formula, the C and C eigenvalues atΓ , M, X are multiplied together. It can be seen that theproduct of determinant of the three representation ma-trices remains unchanged, which confirms that the valueof z C remains stable when the lattice is changed.5 Appendix H: Corner cases of SI
There are 10 corner cases of SIs in MSGs that can-not be induced from the generating SIs. We give themSI formulas written specifically in their coirreps. Theseformulas can be obtained from the Smith normal formdecomposition.
1. Corner cases in type-1 MSGs
MSG 87.75 I /m has indicator group Z , and twonontrivial BSs: b = P − P − X + X b = M + M − M − M (H1)We can calculate z +4 m, , z − m, , z ,S , z (cid:48) P for them, whichequal (2,0,1,1) and (3,1,1,1). It can be seen that z +4 m, , z − m, only form a Z , group, thus cannot be takendirectly as the indicator. However, their combination z +4 m, = 12 ( z +4 m, + z − m, ) mod 4 z − m, = 12 ( z +4 m, − z − m, ) mod 4 (H2)can generate the Z , group, and the two nontrivial BSshave (1,1) and (2,1) of the new indicators. Note that z +4 m, , z − m, in the indicator formula have already mod 4.This MSG has TCI classification Z , whose generatorscan be taken as a translation decoration and a Z deco-ration, which has mirror Chern number (2,0) and (1,-1)respectively, and can be diagnosed by z +4 m, and z − m, .These two indicators are well-defined because they takeinteger values on any combination of the two generators.Note that this MSG does not have Weyl states thatcan be detected by indicator, because it has an extra mir-ror symmetry compared to SG 81 and 82. Weyl pointsalways appear in 8 n number and can be pairwise annihi-lated at generic points on k z = 0 plane without changingirreps at HSPs, while in SG 81 and 82, there are 4 n Weylpoints and can only be annihilated at S invariant HSPs.We remark that there are two type-3 MSGs 139.537 I /mm (cid:48) m (cid:48) and 140.547 I /mc (cid:48) m (cid:48) , whose unitary halvingsubgroup is SG 87, also have these two indicators. Type-4 MSG 87.80 Ic /m has a Z indicator and can be takenas z +4 m, or z − m, , which are equivalent.
2. Corner case in type-2 MSGs
Type-2 MSG 226.123
F m c (cid:48) has indicator group Z ,but the z = 2 z (cid:48) ,S − z (cid:48) P formula introduced in type-1SG P /mmm cannot be used, because some AIs havenon-zero z . As a result, we adopt the SI formula from the Smith normal form: z , . = 3 N (Γ ) + 3 N (Γ ) + 4 N (Γ ) + 2 N (Γ )+4 N ( L L ) − N ( L L ) − N ( X ) + N ( X ) mod 8(H3)One may wonder why the z formula holds in allother MSGs that has Z indicator but only fails inMSG 226.123. This is because 226.123 is the only non-symmorphic MSG with Z indicator, and the inversionand S centers in it cannot coincide.
3. Corner cases in type-3 MSGs
There are 6 corner cases of type-3 MSG whose indi-cators can not adopt the generating SIs. They can bedivided into two classes: five of them have a Z indi-cator and anti-unitary symmetries Glide- T or Mirror- T ,and the other one with an Z indicator group and C · T .These indicators all represent gapped states, as provedlater.The Z class has 5 MSGs: 41.215 Ab (cid:48) a (cid:48)
2, 42.222
F m (cid:48) m (cid:48)
2, 60.424
P b (cid:48) cn (cid:48) , 68.515 Cc (cid:48) c (cid:48) a , and 110.249 I c (cid:48) d (cid:48) . The nontrivial BSs of the corner case Z indi-cator in these MSGs all correspond to topological stateswith even Chern number on the k z = 0 /π plane in theconventional BZ, which cannot be diagnosed using the C Chern number formula. The C Chern number formulawith an extra also fails because it gives nonzero valueswhen applied to some AIs. As a result, we adopt theSI formula from the Smith normal form in these MSGs.Note that 60.424 and 68.515 have indicator group Z , ,with one of the Z being z (cid:48) P . • MSG 41.215 Ab (cid:48) a (cid:48) z , . = N (Γ ) mod 2 (H4) • MSG 42.222
F m (cid:48) m (cid:48) z , . = N (Γ ) − N ( A ) mod 2 (H5) • MSG 60.424
P b (cid:48) cn (cid:48) z , . = N (Γ ) mod 2 (H6) • MSG 68.515 Cc (cid:48) c (cid:48) az , . = N ( Z Z ) − N ( C C ) mod 2 (H7) • MSG 110.249 I c (cid:48) d (cid:48) z , . = N ( M M ) mod 2 (H8)The Z class has only one MSG: 135.487 P (cid:48) /mbc (cid:48) .The nontrivial BS in this MSG has z P = 2 , z m,k z =(1 , , , Z decoration. Notethat z (cid:48) P = 2 for some AIs, which fails to serve as the in-dicator. We adopt the SI formula from the Smith normalform in this MSG:6 • MSG 135.487 P (cid:48) /mbc (cid:48) z , . = N (Γ ) − N ( R R ) + N ( S ) − N ( T ) mod 4(H9)
4. Corner cases in type-4 MSGs
There are 2 corner cases in type-4 MSGs, i.e., 37.185 C a cc P C nc , which both have a Z indica-tor corresponding to Weyl states, with indicator formu-las: • MSG 37.185 C a cc z , . = N ( R ) + N ( T T ) + N ( Z Z ) mod 2 (H10) • MSG 104.209 P C ncz , . = 2 N ( R R ) mod 2 (H11)For 37.185, the indicator can be interpreted as z = N − R + ( N − T + N − Z ) / ZГ A XRM
FIG. 19. A closed loop in the BZ of MSG 104.209.
We choose the loop Z-A-R that encloses one eighthof the 2D BZ of k z = π in Figure.19. The high-symmetry-line Z-A, A-R and R-Z have glide symme-try G , G y and G x , respectively. Starting from Z ,we choose the two eigenstates of G as the bases: | u ( Z ) (cid:105) = (cid:12)(cid:12) u , + i ( Z ) (cid:11) , | u ( Z ) (cid:105) = | u ( Z ) (cid:105) ∗ , and con-sider the phase they gain after the loop evolution.The two bases gain a phase θ after evolving from Z to A and become (cid:18) e iθ e − iθ (cid:19) (cid:18) | u (cid:105)| u (cid:105) (cid:19) (H12)Thanks to the C T symmetry, the phases two bases gainare conjugate to each other. At three HSPs Z, A, R , weneed to perform a basis transformation, because the threeglide symmetries are not simultaneously diagonal. Forexample, at A , we can use C = (cid:18) − i (1 + √ i ( − √ (cid:19) / (cid:113) − i √ G to G y . However,there are two irreps A and A of SG 104 which have dif-ferent C eigenvalues. For A , C transforms G = i, − i to G y = − i, i , while for A , C transforms G = i, − i to G y = i, − i . As a result, we can define a representationmatrix to distinguish this difference: D A ( C ) = (cid:18) (cid:19) if A (cid:18) (cid:19) if A (H14)Similarly, we can define such representation matrix at R and Z : D R ( C ) = (cid:18) (cid:19) if R R (cid:18) (cid:19) if R R D Z ( C ) = (cid:18) (cid:19) if Z (cid:18) (cid:19) if Z (H15)Assume the phase gained on A-R and R-Z for G y and G x eigenstates are θ and θ , respectively, then the totalphase gained after the loop evolution is: M = D Z ( C ) C − (cid:18) e iθ e − iθ (cid:19) D R ( C ) (cid:18) e iθ e − iθ (cid:19) · D A ( C ) C − (cid:18) e iθ e − iθ (cid:19) (H16)The Berry phase of the loop becomesΦ B = det( M ) = det( D Z ( C ) D R ( C ) D A ( C )) (H17)Note that in type-4 MSG 104.209, there are ex-tra degeneracy, i.e., Z Z and Z Z (same for A ).Therefore the determinant det( D Z/A ( C )) ≡
1, whiledet( D R ( C )) = − R R .As a result,Φ B = − N ( R R ) = 1 mod 2 (H18)and thus z , . = 1 indicates the existence of the Weylstate.
5. Interpretation of the type-3 corner caseindicator formulas
First, we show the indicators in type-3 corner cases donot represent Weyl states. The possible Weyl points con-figurations can be determined by looking into the sym-metry operations of the MSG, with proper rotations andtime-reversal symmetry leave the Weyl point chirality un-changed, while improper operations reverse the chirality.71. MSG 41.215 Ab (cid:48) a (cid:48)
2. This MSG has symmetries { , C z , { M x · T | } , { M y · T | }} , and 4 n Weylpoints can be created or annihilated at genericpoints on the z -axis without changing irreps atHSPs, and thus cannot be diagnosed using SI.2. MSG 42.222 F m (cid:48) m (cid:48)
2. This MSG has symmetries { , C z , M x · T, M y · T } . The Weyl points config-uration is similar to MSG 42.215 and cannot bediagnosed using SI.3. MSG 60.424 P b (cid:48) cn (cid:48) . This MSG has generators {{ C y | } , P, { M x · T |
12 12 }} . 4 n Weyl points canbe created or annihilated at a generic point z = z on the z -axis, with the other 4 n Weyl points at z = − z . SI fails to diagnose them.4. MSG 68.515 Cc (cid:48) c (cid:48) a . This MSG has generators {{ C z | } , P, { M x · T |
12 12 }} . The analysis is sim-ilar to MSG 60.424.5. MSG 110.249 I c (cid:48) d (cid:48) . This MSG has generators {{ C z |
12 14 } , { M x · T | } , { M y · T | }} . Simi-larly, 8 n Weyl points can be created or annihilatedaround a generic point on the z -axis.6. MSG 135.487 P (cid:48) /mbc (cid:48) . This MSG has gener-ators {{ C x |
12 12 } , P, { C z · T | }} . Similar to60.424, 8 n Weyl points can be created or annihi-lated around two generic point z = ± z on the z -axis.As a result, the SIs in these 6 type-3 MSGs all corre-spond to the gapped states. Among them, the SI groupof MSG 41.215, 42.222 and 110.249 is Z , and their TCIclassifications are all Z , generated by the translation dec-oration, which means a nontrivial SI corresponds to allthe odd number copies of the translation decoration.MSG 60.424 and 68.515 have Z , indicator group, withone Z being z (cid:48) P , and their TCI classifications are both Z × Z . Here we calculate the Z , indicators for the TCIclassification generators in these two MSGs. a. MSG 60.424 P b (cid:48) cn (cid:48) In MSG 60.424, there aretwo types of possible LCs, i.e., stacking 2D layers alongthe z or y directions. •
2D layers on z = 0 , . The layer at z = 0 hassymmetries { , P, { C x · T | (
12 12 } , { M x · T | (
12 12 }} ,while the other layer at z = is generated by acting g = { C y | } on the z = 0 layer. Assume the2D layer at z = 0 has a wave function W ( r ) andrepresentation D k x k y ( R ), then the layer at z = has wave function W ( r ) = gW ( r ). We calculatethe representation matrix of P and g for the 3DBloch function: a ( k ) = 1 √ N (cid:88) n e ik z · n W ( r − n ) a ( k ) = 1 √ N (cid:88) n e ik z · n W ( r − n ) (H19) As P · g = { | , , − } g · P , we haveˆ P W ( r − n ) = D k x k y ( P ) W ( r + n )ˆ P W ( r − n ) = D k x k y ( P ) W ( r + n − ⇒ ˆ P a ( k ) = D k x k y ( P ) a ( P k )ˆ P a ( k ) = e − ik z D k x k y ( P ) a ( P k ) ⇒ D k ( P ) = (cid:18) D k x k y ( P ) 00 e − ik z D k x k y ( P ) (cid:19) (H20)As a result, D k ( P ) = D k x k y ( P ) ⊕ D k x k y ( P ) forHSPs on the k y = 0 plane, while D k ( P ) = D k x k y ( P ) ⊕ − D k x k y ( P ) for the k y = π plane.For g , as g = −
1, we haveˆ ga ( k ) = a ( gk )ˆ ga ( k ) = − a ( gk ) ⇒ D k ( g ) = (cid:18) − (cid:19) (H21)The representation matrix of g is off-diagonal,which has zero trace. As a result, the numbersof ± i bands must be the same. •
2D layers on y = 0 , . The layer at y = 0 hassymmetries { , P, { C y | (00 ) } , { M y | (00 ) }} , whilethe other layer at y = is generated by h = { C x · T |
12 12 } . Assume the 2D layer at y = 0 has awave function W ( r ) and representation D k x k z ( R ),then the layer at y = has wave function W ( r ) = hW ( r ). We calculate the representation matrix of P and g = { C y | (00 ) } for the 3D Bloch function.As P · h = { | − , − , } h · P , we haveˆ P a ( k ) = D k x k z ( P ) a ( P k )ˆ P a ( k ) = e − i ( k x + k y ) D k x k z ( P ) a ( P k ) ⇒ D k ( P ) = (cid:18) D k x k z ( P ) 00 e − i ( k x + k y ) D k x k z ( P ) (cid:19) (H22)As a result, D k ( P ) = D k x k z ( P ) ⊕ D k x k z ( P ) forΓ , C , while D k ( P ) = D k x k z ( P ) ⊕ − D k x k z ( P ) for Y, C .For g , as g · h = −{ | − , , } h · P (where the extraminus sign comes from the SU matrix), we haveˆ ga ( k ) = D k x k z ( g ) a ( gk )ˆ ga ( k ) = − e i ( k x − k z ) D ∗ k x k z ( g ) a ( gk ) ⇒ D k ( g ) = (cid:18) D k x k z ( g ) 00 − e i ( k x − k z ) D ∗ k x k z ( g ) (cid:19) (H23)where the complex conjugation is because h is anti-unitary. As a result, D k ( g ) = D k x k z ( g ) ⊕ D k x k z ( g )for Γ , Z , while D k ( g ) = D k x k z ( g ) ⊕ − D k x k z ( g ) for Y, C .MSG 60.424 has the following AIs8 a = A A + 2 B + 2 C C + D D + E E + 2Γ + 2Γ + Y Y + Y Y + Z Z + Z Z a = A A + 2 B + 2 C C + D D + E E + 2Γ + 2Γ + Y Y + Y Y + Z Z + Z Z a = A A + 2 B + 2 C C + D D + E E + 2Γ + 2Γ + Y Y + Y Y + Z Z + Z Z a = A A + 2 B + 2 C C + D D + E E + 2Γ + 2Γ + Y Y + Y Y + Z Z + Z Z a = A A + 2 B + C C + C C + D D + E E + 2Γ + 2Γ + Y Y + Y Y + 2 Z Z a = A A + 2 B + C C + C C + D D + E E + 2Γ + 2Γ + Y Y + Y Y + 2 Z Z (H24)and two nontrivial BSs: b = 2Γ − + Z Z − Z Z b = Γ + Γ − Γ − Γ (H25)and three Wyckoff positions:4 a = { (0 , , , ( 12 , , , (0 , ,
12 ) , ( 12 , ,
12 ) } b = { (0 , , , ( 12 , , , (0 , ,
12 ) , ( 12 , ,
12 ) } c = { (0 , y,
14 ) , ( 12 , − y + 12 ,
34 ) , (0 , − y,
34 ) , ( 12 , y + 12 ,
14 ) } (H26) a , , a , and a , correspond to Wyckoff positions 4 a, b and 4 c , respectively. The two nontrivial BSs b and b have indicators ( z (cid:48) P , z , . ) = (1 , , (1 , a , , , are consistent with the z -directional LC, while a , are consistent with the y -directional LC. It can also be seen that b + a , , − b + a , satisfies the representation matrices of z -directional LC, b + a , − b + a satisfies the y -directional LC, while othercombinations do not satisfy. As a result, the z -directionalLC of odd Chern number layer has indicator (1 , y -directional LC of odd Chern number layer has in-dicator (1 , Decoration weak δ i ( z (cid:48) P , z , . ) BSz-directional LC( Z ) 000 1 (1,1) b +AIy-directional LC(translation) 020 1 (1,0) b +AITABLE X. Correspondence of invariants and indicators inMSG 60.424 b. MSG 68.515 Cc (cid:48) c (cid:48) a MSG 68.515 has a trans-lation decoration and a Z decoration, with the firstone being an LC of C = 1 layers on z = 0 , planes,and the other one being a non-LC. They have invari-ants ( δ w, , δ i ) = (2 , , (0 , , Z , indicatorscan be chosen as ( z (cid:48) P , z , . ), and we know that the δ i = 1 states have z (cid:48) P = 1. Because any two ofthe three decorations can be chosen as the generators(e.g., the one with ( δ w, , δ i ) = (0 ,
1) and the one with( δ w, , δ i ) = (2 , z (cid:48) P , z , . ) = (0 , z , . = 1. Wethen only need to determine the z , . indicator forthe translation decoration.The 2D layer on the z = 0 plane has symmetries { , P, g = { C z | (0 } , { M z | (0 }} , and the layer onthe z = plane can be generated by h = { C x · T | (0
12 12 ) } .Following the derivation in MSG 60.424, with the com-mutation relation P · h = { | , − , − } h · P and g · h = − h · g , the representation matrices of P and g are D k ( P ) = (cid:18) D k x k y ( P ) 00 e − i ( k y + k z ) D k x k y ( P ) (cid:19) = (cid:26) D k x k y ( P ) ⊕ D k x k y ( P ) , Γ , Y, B, AD k x k y ( P ) ⊕ − D k x k y ( P ) , Z, C, D, ED k ( g ) = (cid:18) D k x k y ( g ) 00 − D ∗ k x k y ( g ) (cid:19) = D k x k y ( g ) ⊕ D k x k y ( g )(H27)This MSG has the following four AIs (omitting irrele-vant AIs) and two Z nontrivial BSs: a = 2 A + 2 B + C C + C C + 2 D + 2 E + 2Γ + 2Γ + 2 Y + 2 Y + Z Z + Z Z a = 2 A + 2 B + C C + C C + 2 D + 2 E + 2Γ + 2Γ + 2 Y + 2 Y + Z Z + Z Z a = 2 A + 2 B + C C + C C + 2 D + 2 E + 2Γ + 2Γ + 2 Y + 2 Y + Z Z + Z Z a = 2 A + 2 B + C C + C C + 2 D + 2 E + 2Γ + 2Γ + 2 Y + 2 Y + Z Z + Z Z b = Y + Y − Y − Y b = − + 2Γ + Z Z − Z Z (H28)It can be seen that a , , , are compatible with the LC of 2D trivial layers on z = 0 , planes, and b + a , , − b +9 a , are compatible with the translation decoration, i.e.,placing C = 1 2D layers on z = 0 , planes. Note thatcombinations of AIs with b are not compatible.Two nontrivial BSs b , b have ( z (cid:48) P , z , . ) =(1 , , (1 , , Decoration weak δ i ( z (cid:48) P , z , . ) possible BSTranslation 002 1 (1,1) b +AI Z
000 1 (1,0) b +AITranslation+ Z
002 0 (0,1) b + b +AITABLE XI. Correspondence of invariants and indicators inMSG 68.515 c. MSG 135.487 P (cid:48) /mbc (cid:48) The Z decoration ofMSG 135.487 has C m,k z = (1 , − , , , δ i = 1, and weplot it in Fig.20. It can be seen that this decoration isidentical to the Z decoration of P mmm if we first shiftthe origin point by half of the lattice vector in the x di-rection and then rotate it by π/ z direction, whichmakes the BHZ model still applicable. O FIG. 20. The Z decoration of MSG 135.487. Note we do notplot every edge modes for simplicity, which can be inferredfrom the symmetries. The BHZ model used for describing
P mmm has( z (cid:48) P, , z (cid:48) P, , z (cid:48) P, , z (cid:48) P ) = (0 , , , z (cid:48) P indicator isstable when the origin point is changed to any otherinversion centers. Moreover, rotating the Z decorationby π/ z direction is equivalent to change thebase-centered lattice to a simple(principal) lattice, whichagain does not change the z (cid:48) P indicator. As a result,we conclude that the Z decoration of MSG 135.487also has z (cid:48) P = 3. On the other hand, as the BHZmodel has time-reversal symmetry and all 8 TRIMshave Kramer pair of opposite C z eigenvalues, the C z eigenvalues at TRIMS in the Z decoration mustalso appear in opposite pairs when the time-reversalsymmetry is broken. Based on these observations, wefind a compatible set of coirreps for this Z decoration as(2¯Γ , ¯ S , ¯ S , ¯ T , ¯ T , ¯ Y , ¯ Y , ¯ U , ¯ U , ¯ X , ¯ X , ¯ R ¯ R , ¯ Z ¯ Z ).Substituting into the z , . , we have z , . = 3 (H29)for the Z decoration.0 Appendix I: Weyl semimetals in MSGs
There are two types of Weyl semimetals in MSGs thatcan be diagnosed by indicators: • inter-plane Weyl points: lie between k i = 0 and k i = π plane; • in-plane Weyl points: lie on k i = 0 or π plane.where i denotes the main rotation axis of the MSG. Al-though these Weyl points exist at generic momenta, theircreation and annihilation must happen at HSPs, whichmakes it possible to detect them using indicators. Noteall SIs of type-2 spinful MSG cannot represent gaplessstates, as shown in Ref.[30].
1. Type-1 MSGs
Inter-plane Weyl points exist when the Chern numberon k i = 0 and π plane are different. In type-1 MSGs,inversion and S symmetry can help diagnose inter-planeWeyl points.MSG 2.4 P
1, 147.13 P
3, and 148.17 R z P indicator, and its odd values represent inter-plane Weylstates, as the Chern numbers on k z = 0 /π planes aredifferent.MSG 81.33 P I have z , Weyl indicator,which also indicates inter-plane Weyl states.The Weyl points configuration of MSG 2.4 and 81.33are shown in Fig.21(1)(2).
2. Type-4 MSGs
Type-4 MSGs have anti-unitary half translation
T τ ,which forbids the inter-plane Weyl points indicated byinversion and S symmetry in type-1 MSGs. However,type-4 MSGs can host in-plane Weyl points, thanks tothe C T symmetry with reasons given below. C T symmetry transfers the Berry curvature F ( k ) to − F ( k ) on k z = 0 /π plane, thus the Berry phase alongany loop on k i = 0 plane is quantized to 0 or π , with π -Berry phase denoting an odd number of Weyl points (ofthe same chirality) inside the loop. This π -Berry phasecan be readily calculated using C n eigenvalues at HSPs.There are 8 type-4 MSGs that can host in-plane Weylpoints, all having a Z indicator group: • MSG 3.4 P a
2. We can take the Berry phase formulausing C eigenvalues on k y = 0 plane as the indica-tor. z C = 1 indicates the Berry phase around halfBZ is π , implying an odd number of Weyl points. C y dictates a same number of Weyl points of thesame chirality on the other half k y = 0 BZ. Thesame argument holds for the k y = π plane, where Weyl points have opposite chirality from the k y = 0plane, to make sure the total chirality is zero. TheWeyl points configuration is shown in Fig.21(4). • MSG 30.117 P b nc P C nn
2. Thesetwo MSGs can host 4 n Weyl points on k z = π plane, with Weyl points connected by glide sym-metry G x /G y having opposite chirality, as shownin Fig.21(5). The indicator formula can be takenas the Berry phase formula using C eigenvalueson k z = π plane divided by 2 because the coirrepson k z = π plane are all two-fold degenerate withthe same C eigenvalue. The justification of thisindicator is very similar to the Weyl states in SG27 in Ref.[85]. We remark here that although theWeyl points on k z = π plane have opposite chiral-ity, they cannot annihilate with each other withoutchanging the irreps on HSPs, as they are also con-nected by G x/y T with ( G x/y T ) = −
1, as shown inRef.[92]. On k z = 0 plane we have ( G x/y T ) = 1,which means the Weyl points are not protected. • MSG 37.185 C a cc
2. This MSG can also host 4 n Weyl points on k z = π plane, with Weyl pointsconnected by glide symmetry G x /G y having oppo-site chirality, as shown in Fig.21(6). Its SI takesthe corner case formula z , . . • MSG 75.5 P C P C . These two MSGsare very similar to MSG 3.4, and we can take theBerry phase formula using C eigenvalues on the k z = 0 plane as the indicator, with z = 2 dictatingan odd number of Weyl points in a quarter of the2D BZ, i.e., 4 n Weyl points of the same chirality onthe k z = 0 plane. There are also a same number ofWeyl points of the opposite chirality on the k z = π plane. The Weyl points configuration is shown inFig.21(7). • MSG 81.37 P C
4. This MSG can host 4 n Weylpoints on both k z = 0 /π plane, with Weyl pointsconnected by S symmetry having opposite chiral-ity, as shown in Fig.21(8). It can be seen fromAI and nontrivial BSs that z C,k z =0 = z C,k z = π al-ways holds, which means the Weyl points will ap-pear simultaneously on k z = 0 , π planes. This Weylconfiguration can be detected by Berry phase for-mula using S eigenvalues on the k z = 0 plane, as S = C − on k z = 0 /π plane. The justification ofthis indicator is very similar to the Weyl states inSG 81 in Ref.[85]. Notice the Weyl points of op-posite chirality are connected by S symmetry, andthey can only be annihilated at HSPs on k z = 0 /π plane. • MSG 104.209 P C nc . This MSG can host 8 n Weylpoints on k z = π plane, with Weyl points connectedby glide symmetry G x /G y having opposite chiral-ity, as shown in Fig.21(9). Its SI takes the cornercase formula z , . .1 ГY ZA CB E D (6) 37.185
ZГ YTU SX R ZГ (7) 75.5
A XRM ZГ (8) 81.37
A XRM (9) 104.209
ZГ YTSR (4) 3.4 (5) 30.117
ZГ A XRMГX YU VZ R T (1) 2.4 (2) 81.33
ZГ A XRM ГX YU VZ R T (3) 10.46 z =1, z =1,3 ГX YU VZ R T z =1, z =0,2 ГX YU VZ R T z =0, z =1,3 FIG. 21. Minimal configurations for Weyl states in MSGs, with the blue and orange dots represent Weyl points of oppositemonopole charges, and the red lines denote the main rotation axis. (1) MSG 2.4. (2) MSG 81.33. (3) 10.46. (4) MSG 3.4. (5)30.117 and 34.163. (6) 37.185. (7) 75.5 and 77.17. (8) 81.37. (9) 104.209.Weyl states in type-1 MSGsMSG SI SI formula Weyl type2.4 P P
3, 148.17 R Z × Z ∗ z P = 1 , P
4, 82.39 I Z × Z ∗ z , Weyl = 1 inter-planeWeyl states in type-3 MSGs10.46 P (cid:48) /m (cid:48) P (cid:48) /c (cid:48) Z , × Z ∗ z P, = 1 , z P = 1 , P (cid:48) /m (cid:48) Z , z C = 3 , z P = 1 , P (cid:48) /m (cid:48) , 14.79 P (cid:48) /c (cid:48) C (cid:48) /m (cid:48) , 15.89 C (cid:48) /c (cid:48) Z × Z ∗ z P = 1 , Z × Z ∗ z P = 1 , Z × Z ∗ z , Weyl = 1 inter-planeWeyl states in type-4 MSGs3.4 P a Z z C = 1 in-plane30.117 P b nc
2, 34.163 P C nn Z z (cid:48) C = 1 in-plane37.185 C a cc Z z , . = 1 in-plane75.5 P C
4, 77.17 P C P C Z × Z ∗ z C = 2 in-plane104.209 P C nc Z z , . = 1 in-planeTABLE XII. Summary of Weyl states in MSGs. Z ∗ stands for extra Z n factors.
3. Type-3 MSGs
Most of the Weyl states in type-3 MSGs can be in-duced from type-1 MSGs, calculated by inversion or S eigenvalues, as listed in the following table.However, when the MSG also has C i · T symmetry,the inter-plane Weyl points could be pinned on k i = 0 /π planes and become in-plane Weyl. There are three specialtype-3 MSGs that have two indicators to specify the Weylpoints configuration: • MSG 10.46 P (cid:48) /m (cid:48) and 13.69 P (cid:48) /c (cid:48) . These two MSGs all have C y · T , and inversion can be usedto calculate the Berry phase of half k y = π planeusing z P, , with π -Berry phase corresponding tothe Weyl points of opposite chirality on the k y = π plane. We plot the Weyl point configurations ofMSG 10.46 in Fig.21(3): – If z P, = 1, when (i) z P = 1 ,
3, the Berryphase of half of k y = 0 /π plane differs by π andonly the k y = π plane has Weyl points, while(ii) z P = 0 , k y = 0 /π planeshost in-plane Weyl points.2 – If z P, = 0, when (i) z P = 1 ,
3, only k y =0 plane hosts in-plane Weyl points, while (ii) z P = 0 , P (cid:48) /m (cid:48) , 14.79 P (cid:48) /c (cid:48) ,12.62 C (cid:48) /m (cid:48) , and 15.89 C (cid:48) /c (cid:48) , which also have C T , there is only one z indicator to specify theWeyl points configuration. This is because in thefirst two MSGs the coirreps are two-fold degenerateof opposite parity on the k y = π plane such thatWeyl points can only appear on the k y = 0 plane,while the last two MSGs are C -face-centered lat-tice, with z P = 1 , π -Berry phase of half k y = 0 plane of the conventionalBZ and corresponding to an odd number of Weylpoints on it. • MSG 175.141 P (cid:48) /m (cid:48) . S symmetry in this MSGcan be used to calculate the π -Berry phase of onesixth of the k z = 0 plane, as S = C − on k z = 0plane. k = 0 /π planes can both host in-plane Weylpoints, with Weyl points connected by S symme-try having opposite chirality. – If z C = 3, when (i) z P = 1 ,
3, the Berryphase of half of k z = 0 /π plane differs by π andonly the k z = π plane has Weyl points, while(ii) z P = 0 , k z = 0 /π planeshost in-plane Weyl points. – If z C = 0, when (i) z P = 1 ,
3, only k z =0 plane hosts in-plane Weyl points, while (ii) z P = 0 , Appendix J: Summary of generating SIs in MSGs
We summarize the generating SI formulas in this sec-tion. The first subscript number in the SI name repre-sents its order, and the prime in some indicators repre-sents they are of the original indicators without prime.
1. Generating SIs in type-1 & 2 MSGs • MSG 2.4 P z P,i = (cid:88) k ∈ TRIM ,k i = π
12 ( N − k − N + k ) mod 2 z P = (cid:88) k ∈ TRIM
12 ( N − k − N + k ) mod 4 (J1)where N ± k is the number of valence bands havingpositive (negative) parity. • MSG 11.50 P /m . z (cid:48) P = 12 z P = (cid:88) k ∈ TRIM
14 ( N − k − N + k ) mod 2 (J2)Note this SI is not included as a generating SI inthe first section, because it can be directly inducedfrom z P by taking only even values. We list it herefor ease of consulting. • MSG 47.249
P mmm . z (cid:48) P,i = (cid:88) k ∈ TRIM ,k i = π
14 ( N − k − N + k ) mod 2 z (cid:48) P = (cid:88) k ∈ TRIM
14 ( N − k − N + k ) mod 4 (J3) • Indicators defined by C n rotations:MSG 3.1 P z C = (cid:88) l ∈ occ ln( B lC (Γ) B lC ( X ) B lC ( Y ) B lC ( M )) / ( iπ ) mod 2= (cid:88) k ∈ Γ ,M,X,Y
12 ( N − k − N + k ) mod 2 (J4)MSG 75.1 P z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( M ) B lC ( Y )) / ( i π P z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( K ) B lC ( K (cid:48) )) / ( i π P z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( K ) B lC ( M )) / ( i π C stands for Chern number or C n rotation,and B iC n ( k ) represents the C n eigenvalue of the i -thband at HSP k . This Chern number is calculatedon the k i = 0 plane by default, and will be changedto z nC,π if calculated on the k i = π plane.Note that these SIs are also adopted in some otherMSGs and take only even values, for example, the z C can only take 0 , I
4, and we denotethem as z nC / • Indicators defined by C n rotations and mirror sym-metry to calculate the mirror Chern numbers modn, which can be calculated by z nC formula using M = ± i bands on k i = 0 /π plane. We name theseindicators as z ± nm, /π , where 0 /π denotes which k plane and ± which mirror sector.MSG 10.41 P /m : z +2 m, , z − m, , z +2 m,π .MSG 83.43 P /m : z +4 m, , z − m, , z +4 m,π .MSG 174.133 P P /m : z +3 m, , z − m, , z +3 m,π .MSG 175.137 P /m : z +6 m, , z − m, , z +6 m,π . • MSG 81.33 P
4. In this MSG, aside from z C , wecan also define two Z indicators z ,S and z , Weyl to represent the S decoration and inter-plane Weylstates: µ S = 1 √ (cid:88) i ∈ occ. (cid:88) k ∈ K S β i ( k ) z ,S = 12 (Re µ S − Im µ S ) mod 2 z , Weyl = Re( µ S ) mod 2 (J8)where β i ( k ) = e α π i ( α = 1 , , ,
7) are the S eigen-values at S -invariant points K S . • MSG 81.34 P (cid:48) . This type-2 MSG has the follow-ing z (cid:48) ,S Z indicator: z (cid:48) ,S = − µ S = (cid:88) k ∈ K S
12 ( n −√ k − n √ k ) (J9)where n ±√ k is the number of Kramer pairs at S -invariant TRIMs with tr [ D ( S )] = ±√ z (cid:48) ,S = 1represents a STI. • MSG 123.339 P /mmm . This MSG has an Z in-dicator which is taken as z = 2 z (cid:48) ,S − z (cid:48) P mod 8 (J10)Note that z (cid:48) P and z (cid:48) ,S in z formula should nottake mod, and their original values are used to cal-culate z .4 • MSG 191.233 P /mmm . This MSG has an Z indicator which be taken as z = { z m,S + 3 [( z m,S − z (cid:48) P ) mod 4] } mod 12(J11)where z m,S has more than one choices, and herewe choose it as z m,S = z +6 m, + z +6 m,π .In type-2 MSGs, there are similar z indicators,where in MSG 175.138, 191.234 and 192.244 thedefinition of z m,S is the same, while in MSG176.144, 193.254 and 194.264, z m,S = z m, be-cause they have C -screw which forces the mirrorChern number on k z = π plane to be zero.
2. Generating SIs in type-3 MSG • MSG 27.81
P c (cid:48) c (cid:48) z (cid:48) C = 12 z C mod 2 (J12)MSG 103.199 P c (cid:48) c (cid:48) z (cid:48) C = 12 z C mod 4 (J13)MSG 184.195 P c (cid:48) c (cid:48) . z (cid:48) C = 12 z C mod 6 (J14)where z nC on the RHS has not taken mod.
3. Corner cases SI
Type-1 MSG: • I /mz +4 m, = 12 ( z +4 m, + z − m, ) mod 4 z − m, = 12 ( z +4 m, − z − m, ) mod 4 (J15)Type-2 MSG • F m c (cid:48) z , . = 3 N (Γ ) + 3 N (Γ ) + 4 N (Γ ) + 2 N (Γ )+4 N ( L L ) − N ( L L ) − N ( X ) + N ( X ) mod 8(J16)Type-3 MSGs: • MSG 41.215 Ab (cid:48) a (cid:48) z , . = N (Γ ) mod 2 (J17) • MSG 42.222
F m (cid:48) m (cid:48) z , . = N (Γ ) − N ( A ) mod 2 (J18) • MSG 60.424
P b (cid:48) cn (cid:48) z , . = N (Γ ) mod 2 (J19) • MSG 68.515 Cc (cid:48) c (cid:48) az , . = N ( Z Z ) − N ( C C ) mod 2 (J20) • MSG 110.249 I c (cid:48) d (cid:48) z , . = N ( M M ) mod 2 (J21) • MSG 135.487 P (cid:48) /mbc (cid:48) z , . = N (Γ ) − N ( R R ) + N ( S ) − N ( T ) mod 4(J22)Type-4 MSGs (both are Weyl states): • MSG 37.185 C a cc z , . = N ( R ) + N ( T T ) + N ( Z Z ) mod 2 (J23) • MSG 104.209 P C ncz , . = 2 N ( R R ) mod 2 (J24)5 Appendix K: The Berry phase of closed loops in 2DBZ
ГY X ГM Y XMГ K K ’ K ’’ Г KMM ’ (a) C (b) C (c) C (d) C FIG. 22. Loops and HSPs used to calculate the Berry phaseon 2D BZs
The formulas for calculating the Berry phase Φ B around a closed loop in Fig.22 using C n eigenvalues arederived in Ref.[49]: C : e i Φ B = (cid:89) l ∈ occ . B lC (Γ) B lC ( X ) B lC ( Y ) B lC ( M ) C : e i Φ B = (cid:89) l ∈ occ . ( − B lC (Γ) B lC ( M ) B lC ( Y ) C : e i Φ B = (cid:89) l ∈ occ . ( − B lC (Γ) B lC ( K ) B lC ( K (cid:48) ) C : e i Φ B = (cid:89) l ∈ occ . ( − B lC (Γ) B lC ( K ) B lC ( M ) (K1)where B lC n ( k ) represent the C n eigenvalue of the l -thband at C n -invariant point k .These formulas can be used to diagnose Weyl points.For MSGs with P T symmetry, Berry curvature is forcedto be zero (for gapped momenta). Also for MSGs with C i T , the Berry curvature on k i = 0 , π planes are forcedto be zero (for gapped momenta). As a result, in theseMSGs, the Berry phase around any closed loops equalszero, or equals π when there are an odd number of Weylpoints lie in these loops. These formulas using C n eigen-values can be used to diagnose the π Berry phase.6
Appendix L: The BHZ model for describingnon-layer constructions
Bernevig-Hughes-Zhang(BHZ) model was first intro-duced in the 2D HgTe system to characterize thequantum-spin Hall effect (QSHE), which can be gener-alized to 3D to describe strong TIs. In this appendix, wecalculate the mirror Chern numbers and SIs of the BHZmodel.
1. 2D Chern insulator model
We first introduce a 2D Chern insulator model, whichis known as the half-BHZ model or the Qi-Wu-Zhangmodel[93]: H = H ( k ) · σ H x ( k ) = sin( k x ) H y ( k ) = sin( k y ) H z ( k ) = M − cos( k x ) − cos( k y ) (L1)which has eigenvalues E ± = ±| H ( k ) | . The eigenvectorof E − has two different forms u − ( k ) = 1 N − (cid:18) H z − | H | H x + iH y (cid:19) ,u − ( k ) = 1 N − (cid:18) − H x + iH y H z + | H | (cid:19) (L2)where N , − are normalization factors. These two eigen-vectors differ by a phase u − = u − e iφ ⇒ e iφ = | H x + iH y | H x + iH y ≈ | k x + ik y | k x + ik y = e − iθ (L3)where we use the approximation sin( k ) ≈ k if k → e iθ = ( k x + ik y ) /k . The Berry connection satisfies A − = A − + ∇ k φ . When H x = H y = 0, u − = 0 if H z > u − = 0 if H z <
0. As a result, we need to choose u , − properly depending on whether H z ( k ) is positive ornegative.The upper band also has two forms of eigenvectors u ( k ) = 1 N (cid:18) H z + | H | H x + iH y (cid:19) ,u ( k ) = 1 N (cid:18) − H x + iH y H z − | H | (cid:19) (L4)These two eigenvectors differ by a phase u = u e iφ ⇒ e iφ = − H x + iH y | − H x + iH y | ≈ − k x + ik y | − k x + ik y | = e − iθ (L5) (0,�) (�,0)(�,�) (a) 0 When H x = H y = 0, u = 0 if H z < 0, while u = 0 if H z > 0. Note that the choice of u , is opposite to u , − ,which ensures C + = − C − . The zero total Chern numberis also enforced by the time-reversal symmetry (TRS).The topological property of the this 2D BHZ modeldepends on the sole parameter M . If M = 0 , ± 2, themodel is gapless, which are topological phase transitionpoints. If M > M < − 2, the model is equivalentto M → ±∞ , which is trivial atomic insulator. If 0 2, the model is topological and has C ± = ∓ 1, where C ± is the Chern number of E ± band, while C ± = ± − < M < < M < 2, we have H z < k = (0 , H z > k = ( π, , (0 , π ) , ( π, π ). We can choose u − in the vicinity of Γ and u − in other momenta. The Chernnumber C − can be calculated as2 πC − = (cid:90) (cid:90) D ∇ k × A − + (cid:90) (cid:90) D ∇ k × A − = (cid:90) ∂D A − + (cid:90) ∂D A − = (cid:90) ∂D ( A − − A − )= (cid:90) ∂D ( −∇ k φ ) = − ( φ (2 π ) − φ (0))= 2 π ⇒ C − = 1 (L6)If − < M < 0, we have H z < k =(0 , , ( π, , (0 , π ) and H z > k = ( π, π ). As aresult, we can choose u − in the vicinity of k = ( π, π )and u − in other momenta. The phase e iφ = e iθ near k = ( π, π ), where e iθ = ( k x − π + i ( k y − π )) / | k − ( π, π ) | .The Chern number C − = − ∂D . a. When H z adds a minus sign Here we summarizethe result of 2D BHZ model when H z adds a minus sign,i.e., H z = − ( M − cos( k x ) − cos( k y )) (L7)In this case, the eigenvalues and the form of eigenvectorsare not changed, but the choice of eigenvectors change:7 • If 0 < M < 2, we have H z > k = (0 , H z < k = ( π, , (0 , π ) , ( π, π ). Wecan choose u − in the vicinity of Γ and u − in othermomenta. As a result, we have C ± = ± • If − < M < 0, we have H z > k =(0 , , ( π, , (0 , π ) and H z < k = ( π, π ).As a result, we can choose u − in the vicinity of k = ( π, π ) and u − in other momenta. As a result,we have C ± = ∓ 2. 3D BHZ model A simplified version of 3D BHZ model can be writtenas H ( k ) = ( M − (cid:88) i =1 , , cos( k i ))Σ + (cid:88) i =1 , , sin( k i )Σ i (L8)where Σ ij = σ i ⊗ σ j and M is the sole parameter.This model has O h and time-reversal symmetry. As-sume a symmetry operation R has representation matrix D ( R ). The Hamiltonian with symmetry D ( R ) satisfies H ( k ) = D − ( R ) H ( Rk ) D ( R ). We can calculate the rep-resentation matrix for the following symmetries: • Inversion P : D ( P ) = Σ . • M x : D ( M x ) = i Σ . • M y : D ( M y ) = i Σ . • M z : D ( M z ) = i Σ . • D ( M ): D ( M ) = i √ (Σ − Σ ). • S : D ( S ) = √ (Σ − i Σ ). • T : D ( T ) = i Σ ˆ K , where ˆ K is the complex conju-gation.Here we mainly discuss the M − ∈ (0 , 2) phase andcalculate the mirror Chern numbers along with SI. The M − ∈ ( − , 0) phase can be derived similarly. a. Mirror Chern number on k z = 0 , π planes When k z = 0, the Hamiltonian reduces to H ( k z = 0) =( M − − cos( k x ) − cos( k y ))Σ + sin( k x )Σ + sin( k y )Σ (L9)We can block-diagonalize H ( k ) using S = (L10) ⇒ H (cid:48) ( k z = 0) = S − H ( k z = 0) S = [sin( k x ) σ + sin( k y ) σ + M ( k ) σ ] ⊕ [sin( k x ) σ + sin( k y ) σ − M ( k ) σ ](L11) where M ( k ) = M − − cos( k x ) − cos( k y ). The represen-tation matrix of P and M z becomes D (cid:48) ( P ) = S − D ( P ) S = − − ,D (cid:48) ( M z ) = S − D ( M z ) S = i i − i 00 0 0 − i (L12)We arrange the eigenvalues as E = ( E + ( k ) , E − ( k )) ⊕ ( E + ( k ) , E − ( k )), and calculate the corresponding eigen-vectors at different TRIMs. At Γ, we have | ψ Γ (cid:105) = ( u (Γ) , u − (Γ)) ⊕ ( u (Γ) , u − (Γ)) = (L13)Under this basis, the representation matrix of P and M z becomes D (cid:48)(cid:48) ( P ) = (cid:104) ψ Γ | D (cid:48) ( P ) | ψ Γ (cid:105) = − − ,D (cid:48)(cid:48) ( M z ) = (cid:104) ψ Γ | D (cid:48) ( M z ) | ψ Γ (cid:105) = i i − i 00 0 0 − i (L14)As a result, we conclude that the E − (Γ) bands is aKramers pair with inversion eigenvalue +1, while the E + (Γ) bands a another with inversion eigenvalue − k = ( π, , , (0 , π, , ( π, π, | ψ k (cid:105) = ( u ( k ) , u − ( k )) ⊕ ( u ( k ) , u − ( k )) = (L15)Under this basis, the representation matrix of P and M z becomes D (cid:48)(cid:48) ( P ) = (cid:104) ψ k | D (cid:48) ( P ) | ψ k (cid:105) = − − ,D (cid:48)(cid:48) ( M z ) = (cid:104) ψ k | D (cid:48) ( M z ) | ψ k (cid:105) = i i − i 00 0 0 − i (L16)The inversion eigenvalues of these three TRIMs are oppo-site from Γ, i.e., E − ( k ) band being − E + ( k ) bandbeing +1. The E − (Γ) bands form a Kramers pair whose8mirror Chern number are C + m,k z =0 = 1 , C − m,k z =0 = − E + (Γ) bands form another Kramers pair whosemirror Chern numbers are C + m,k z =0 = − , C − m,k z =0 = 1On k z = π plane, the Hamiltonian can be block-diagonalized similarly, with M ( k ) = M + 1 − cos( k x ) − cos( k y ). M − ∈ (0 , 2) implies that M + 1 > 2, i.e., themodel is in the trivial phase, and we can choose the sameeigenvector on 4 TRIMs on k z = π plane: | ψ k (cid:105) = ( u ( k ) , u − ( k )) ⊕ ( u ( k ) , u − ( k )) = (L17)which is the same as the three TRIMs on k z = 0 planeexcept Γ. As a result, we have the parity of E − ( k ) bandbeing − E + ( k ) band being +1. The mirror Chernnumbers are both zero for M z = ± i sectors.From the analysis above, we conclude that when M ∈ (1 , E − has mirror Chern number( C + m, , C − m,π , C + m,π , C − m,π ) = (1 , − , , − z (cid:48) P = 3 (L18)The mirror Chern numbers in the k x and k y direc-tion are the same as k z direction as the model has C symmetry in the (111) direction, which makes the threedirections equivalent. b. Mirror Chern number in the (1 , ¯1 , direction. For the O h group, mirror x, y, z are in the same con-jugacy class, while mirror (1 , ± , , (1 , , ± , (0 , , ± M symmetry: M = . The represen-tation matrix of M satisfiesΣ D ( M ) = D ( M )Σ , Σ D ( M ) = D ( M )Σ , Σ D ( M ) = D ( M )Σ , Σ D ( M ) = D ( M )Σ , ⇒ D ( M ) = i √ − Σ ) (L19)We can diagonalize it using: S = 1 √ √ i − i √ √ i − i √ ⇒ D (cid:48) ( M ) = S − D ( M ) S = i − i − i 00 0 0 i (L20) The k = 0 plane is defined by k x = k y . On thisplane, the Hamiltonian becomes H ( k = 0) = ( M − k x ) − cos( k z ))Σ +sin( k x )(Σ + Σ ) + sin k z Σ (L21)We first change basis using S , where the Hamiltonianbecomes very similar to H ( k z = 0), and them we block-diagonalize it using S : H (cid:48) ( k = 0) = ( S S ) − H ( k = 0) S S = [ t σ + t σ + M ( k ) σ ] ⊕ [ t σ + t σ − M ( k ) σ ] (L22)where t = sin( k x ) − √ sin k z , t = sin( k x ) + √ sin k z , M ( k ) = M − k x ) − cos( k z ). The repre-sentation matrix of P and M becomes D (cid:48) ( P ) = ( S S ) − D ( P ) S S = − − ,D (cid:48)(cid:48) ( M ) = S − D (cid:48) ( M ) S = i i − i 00 0 0 − i (L23)The mirror Chern number can be calculated similarlyas C m, = ( − , t and t . c. S eigenvalues at S -invariant points The S symmetry in z direction has O (3) rotation matrix S = − − . The representation matrix of S satisfiesΣ D ( S ) = D ( S )Σ , Σ D ( S ) = − D ( S )Σ , Σ D ( S ) = D ( S )Σ , Σ D ( S ) = − D ( S )Σ , ⇒ D ( S ) = 1 √ − i Σ ) (L24)There are four S -invariant momenta Γ, M ( π, π, X (0 , π, R ( π, π, π ). Using the eigenvectors de-rived before, we have the corresponding representationmatrices: D (cid:48) Γ ( S ) = (cid:104) ψ Γ | S − D ( S ) S | ψ Γ (cid:105) = 1 √ − − i − i − i 00 0 0 1 + i D (cid:48) M,X,R ( S ) = (cid:104) ψ M,X,R | S − D ( S ) S | ψ M,X,R (cid:105) = 1 √ − i − − i i 00 0 0 − i (L25)9As a result, the S eigenvalue of the E − Kramer pair is √ −√ M, X, R . The S indicator can becalculated as z (cid:48) ,S = (cid:88) k ∈ K S 12 ( n −√ k − n √ k ) = 1 (L26)The z indicator can be calculated as z = 2 z (cid:48) ,S − z (cid:48) P mod 8 = 7 (L27) d. z indicator for the Z decoration in MSG P m ¯3 m The Z decoration in MSG 221.92 P m ¯3 m has mir-ror Chern number C m,k z = (1 , − , , 0) and C m, =(1 , − M ∈ (1 , C m,k z = (1 , − , , C m, = ( − , , C m,k z = (0 , , , C m, = (2 , − 2) and z = 4. As a result, the Z decora-tion has z = 3. e. Verification of the z C formula For the BHZmodel, we can calculate its C and C eigenvalues onHSPs. When 1 < M < 3, as C = S · P, C = M z · P ,we have D Γ ( C ) = 1 √ i − i − i 00 0 0 1 + i ,D M ( C ) = 1 √ − i i i 00 0 0 1 − i D X ( C ) = i − i − i 00 0 0 i (L28)For the lower band with M z = + i , we have B C (Γ) = e − iπ/ , B C ( M ) = e iπ/ , B C ( X ) = − i , which leads to z C = (cid:88) l ∈ occ ln( − B lC (Γ) B lC ( M ) B lC ( Y )) / ( i π Appendix M: The construction of atomic insulatorbasis in MSGs In this Appendix, following Ref.[94], we show theprocedures for constructing the band representation ofatomic insulators in SGs and MSGs. 1. General procedures a. Wyckoff positions Wyckoff positions (WPs) arespecific points in the unit cell of an (M)SG G , classifiedby their site symmetry groups (SSGs). The SSG of agiven WP q is defined as the MSG elements that keep q fixed, i.e. G q = { g ∈ G | g q = q } Caution that here we require that g q = q , but not themore usual g q = q + R , where R is a lattice vector. Thisconvention is more convenient as the symmetry opera-tions will not change the center of Wannier functions toanother unit cell.The representatives g α of the quotient group G/G q areused to generate the equivalent positions of q : q α = g α q , g α ∈ G/G q where g α may also contain lattice translations s.t q α lo-cates in the home cell. The SSG of q α is G q α ≡ (cid:8) g α hg − α | h ∈ G q (cid:9) b. Wannier functions Assume a given set of equiv-alent WPs { q , q , ... q n } has SSG G q . We can place lo-calized Wannier functions W i ( r ) on q , which form thebases of an m-dimensional representation D ij ( g ) of G q : gW i ( r ) = W i ( g − r ) = (cid:88) j W j ( r ) D ji ( g ) , g ∈ G q (M1)where i = 1 , ..., m indicate m bases, and the second indexindicates n sites. Notice that g ∈ G q may contain someinteger translations s.t W i ( g − r ) lies in the same unitcell as W i ( r ), and in more general case, W i ( g − r ) = (cid:80) j W j ( r − R µ ) D ji ( g ), where R µ = g q − q .The Wannier functions on q α are W iα ( r ) = g α W i ( r ) = W i ( g − α r ) (M2)which form the bases of the conjugate representation D α ( g ) = D ( g − α gg α ).Wannier functions in other unit cells transform simi-larly: gW i ( r − R µ ) = g { | R µ } W i ( r ) = { | gR µ } gW i ( r )= { | gR µ } W i ( g − r )= (cid:88) j W j ( r − gR µ ) D ji ( g ) , g ∈ G q (M3) For other symmetry operations h / ∈ G q , we can derivethe following relation using coset representatives: hg α = { | R βα } g β g, R βα = h q α − q β (M4)where g α , g β are coset representatives and g ∈ G q . Forgiven h and g α , we can find g β by enumerating cosetrepresentatives s.t. g β hg α ∈ G q . The translation R βα exists because we set Wannier functions W iα in the sameunit cell and can be derived as: hg α q = h q α = { | R βα } g β g q = { | R βα } g β q = { | R βα } q β ⇒ R βα = h q α − q β (M5)where we have used g q = q , g i q = q i . These relationshold if we take the convention that integer translationsare absorbed into g, g i s.t. q , q i lies in the home cell. Inmore general cases, R βα = t hg α − g β g (M6)where t denotes the translation part of the operation.With Eq.M4, we consider how W iα transform underarbitrary h = { R | τ } / ∈ G q , i.e., lifting the representationof G q to a representation of G : hW iα ( r − R µ ) = h { | R µ } g α W i ( r ) = { | RR µ } hg α W i ( r )= { | RR µ + R βα } g β gW i ( r )= { | RR µ + R βα } (cid:88) j W jβ ( r ) D ji ( g )= (cid:88) j W jβ ( r − RR µ − R βα ) D ji ( g ) (M7) c. Bloch functions The Bloch functions are gener-ated from the Wannier functions: a iα ( k, r ) = 1 √ N (cid:88) µ e ik · R µ W iα ( r − R µ ) (M8)For h ∈ G , they transform as ha iα ( k, r ) = 1 √ N (cid:88) µ e ik · R µ hW iα ( r − R µ )= 1 √ N (cid:88) µ e ik · R µ (cid:88) j W jβ ( r − RR µ − R βα ) D ji ( g )= e − iRk · R βα (cid:88) j D ji ( g )1 √ N (cid:88) µ e iRk · ( RR µ + R βα ) W jβ ( r − ( RR µ + R βα ))= e − iRk · R βα (cid:88) j D ji ( g ) a jβ ( Rk, r ) (M9)As a result, the band representation D kG ( h ) is: D kG ( h ) jβ,iα = (cid:104) a jβ ( Rk, r ) | h | a iα ( k, r ) (cid:105) = e − iRk · R βα D ji ( g ) (M10)where g = g − β { | − R βα } hg α , R βα = t hg α − g β g .1 d. Note: non-orthogonal rotations in trigonal andhexagonal lattices The formula above failed in trigonaland hexagonal lattice, because the 3D rotation matricesare not orthogonal, i.e., R T (cid:54) = R − ⇒ k · R µ (cid:54) = Rk · RR µ .Instead, ( k, R µ ) = k T · R µ = k T · R − RR µ = k T R − · RR µ = (( R − ) T k, RR µ ). As a result, we have D kG ( h ) jβ,iα = (cid:104) a jβ ( kR − , r ) | h | a iα ( k, r ) (cid:105) = e − ikR − · R βα D ji ( g ) (M11) e. Note: Bilbao irrep convention D kG is a “small rep-resentation” when restricted to the little group G k = { h = { R | τ } ∈ G | R k = k } , as the representation matri-ces of two operations that differ by a lattice translationdiffer by a phase factor e − iRk · R µ = e − ik · R µ .However, Bilbao irrep matrices take the conventionthat a lattice translation is represented by e ik · R µ . As aresult, if we want to decompose the band representation D kG into Bilbao irreps, we need to take the conventionthat D kG ( h ) jβ,iα = e ikR − · R βα D ji ( g ) (M12)This is legitimate because it is equivalent to modify therule of action gW ( r ) = W ( g − r ) to gW ( r ) = W ( gr ). f. Decompose into small representations For a fixedmomenta k , D kG ( h ) is a nm × nm -dimensional matrix andcan be block-diagonalized into the small representationsof the little group G k :( D ↑ G ) ↓ G k ∼ = (cid:77) i m k i σ k i (M13)where σ k i are irreps of G k .The characters of D kG ( h ) are χ kG ( h ) = (cid:88) i,α D kG ( h ) iα,iα = (cid:88) α e − iRk · R αα χ [ D ( g )](M14)The decomposition coefficients m k i can be readily calcu-lated using the orthogonal theorem of characters. g. Summary of the procedures for deriving AI Wesummarize here the general procedure for deriving theBS of AIs:1. Choose a Wyckoff point q and find the coset de-composition G = (cid:83) α =1 g α G q .2. Choose an irrep D of G q . For a given HSP k , cal-cuate the character χ kG ( h ):(a) for each h ∈ G k and coset representative g α ,find g = g − α { | − R αα } hg α (b) calculate the summation χ kG ( h ) = (cid:80) α e ikR − · R αα ˜ χ [ D ( g )].3. Use the orthogonal theorem of characters to decom-pose χ kG ( h ) into the irreps of G k : χ k G ( h ) = (cid:88) i m k i χ k σ i ( h ) 2. Modification of the procedures for anti-unitarysymmetries In the above derivation, we have only considered uni-tary symmetries, i.e., the derived AIs belong to type-1MSG. For anti-unitary symmetries, we make the follow-ing modification:1. In the relation hg α = { | R βα } g β g, ⇒ g = g − β { | − R βα } hg α (M15)the ± g needto consider complex conjugation if the symmetriesare anti-unitary.2. When g β is anti-unitary, D ji ( g ) needs to take com-plex conjugation: hW iα ( r − R µ ) = h { | R µ } g α W i ( r ) = { | RR µ } hg α W i ( r )= { | RR µ + R βα } g β gW i ( r )= { | RR µ + R βα } (cid:88) j W jβ ( r ) D ∗ ji ( g )= (cid:88) j W jβ ( r − RR µ − R βα ) D ∗ ji ( g ) (M16)3. When g β is anti-unitary ( h unitary), the Blochfunction transforms as: ha iα ( k, r ) = 1 √ N (cid:88) µ e ik · R µ hW iα ( r − R µ )= 1 √ N (cid:88) µ e ik · R µ (cid:88) j W jβ ( r − RR µ − R βα ) D ∗ ji ( g )= e − iRk · R βα (cid:88) j D ∗ ji ( g )1 √ N (cid:88) µ e iRk · ( RR µ + R βα ) W jβ ( r − ( RR µ + R βα ))= e − iRk · R βα (cid:88) j D ∗ ji ( g ) a jβ ( − Rk, r ) (M17)4. When g β is anti-unitary, the band representationbecomes: D kG ( h ) jβ,iα = (cid:104) a jβ ( − Rk, r ) | h | a iα ( k, r ) (cid:105) = e − iRk · R βα D ∗ ji ( g ) (M18)To sum up, under Bilbao convention, the band repre-sentation D kG is D kG ( h ) jβ,iα = e + iRk · R βα D ji ( g ) , h ∈ G, g β unitary D kG ( h ) jβ,iα = e + iRk · R βα D ∗ ji ( g ) , h ∈ G, g β anti-unitary(M19)2 Appendix N: Table: TCI classifications of MSGs TABLE XIII: TCI classifications of MSGs1.1 Z Z Z × Z Z Z Z × Z Z Z Z Z Z × Z Z Z Z Z Z × Z Z Z Z M × Z Z Z M Z M Z M Z × Z Z Z Z Z Z Z Z M × Z Z Z M Z M Z × Z Z Z Z Z M × Z Z M Z × Z Z M Z M Z M Z M × Z Z M Z × Z Z M Z M Z M Z M × Z Z M Z × Z Z M Z M Z × Z Z Z × Z Z Z Z Z Z Z × Z Z Z × Z Z Z Z Z Z Z × Z Z Z × Z Z Z Z × Z Z Z Z Z × Z Z × Z Z Z Z Z Z Z × Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z × Z Z × Z Z Z Z Z × Z Z × Z Z Z Z Z × Z Z Z × Z Z Z × Z Z Z M Z M × Z Z Z M Z M Z M Z M Z M Z M Z × Z Z M × Z Z Z M Z M Z M Z M Z M Z M Z M Z Z × Z Z Z Z Z Z Z Z M Z × Z Z M × Z Z Z M Z M Z M Z M Z M Z M Z M Z Z × Z Z × Z Z Z Z Z Z Z Z Z Z Z × Z Z × Z Z Z Z Z Z Z Z Z Z M Z × Z Z M × Z Z Z M Z M Z M Z M Z M Z M Z M Z Z × Z Z Z Z Z Z Z Z Z × Z Z × Z Z Z Z Z Z Z Z Z Z Z × Z Z Z Z Z Z Z Z M Z M × Z Z Z M Z M Z M Z M Z × Z Z M × Z Z Z M Z M Z M Z Z × Z Z Z Z Z M Z M Z M × Z Z M × Z Z Z M Z M Z M Z M Z M × Z Z × Z Z Z M Z M Z Z M Z × Z Z M × Z Z Z M Z M Z M Z Z × Z Z × Z Z Z Z Z Z M Z M × Z Z Z M Z Z × Z Z Z Z M Z M × Z Z Z M Z M Z Z × Z Z Z Z Z M Z × Z Z M × Z Z Z M Z M Z M Z M Z M Z M × Z Z M Z M Z M Z Z Z × Z Z Z Z Z M Z M Z Z M × Z Z × Z Z M Z M Z M Z M Z M Z Z Z Z × Z Z × Z Z Z Z Z Z Z M Z M Z M Z M Z × Z Z M × Z Z M × Z Z M Z M Z M Z M Z M Z M Z M Z Z Z Z Z × Z Z × Z Z × Z Z Z Z Z Z Z Z Z M Z Z M Z M Z × Z Z M × Z Z × Z Z M Z M Z M Z M Z M Z M Z M Z Z Z Z Z × Z Z × Z Z × Z Z Z Z Z Z Z Z Z M Z M Z Z M × Z Z × Z Z M Z M Z M Z M Z M Z Z Z Z × Z Z × Z Z Z Z Z Z Z M Z M Z M Z Z M × Z Z × Z Z × Z Z M Z M Z M Z M Z M Z M Z M Z M Z M Z Z M × Z Z × Z Z M Z M Z M Z M Z M Z M Z M Z M Z × Z Z M × Z Z M Z M Z M Z M Z M Z Z Z Z Z × Z Z × Z Z × Z Z Z Z Z Z Z Z Z Z Z × Z Z Z Z Z M Z M Z Z M Z × Z Z × Z Z M × Z Z M Z M Z M Z M Z M Z M Z M Z M Z M Z M Z M Z M × Z Z M × Z Z × Z Z M Z M Z M Z M Z Z M Z M Z × Z Z M × Z Z × Z Z M Z M Z M Z M Z M Z M Z M × Z Z M × Z Z M Z M Z M Z M Z M Z Z M × Z Z × Z Z M Z M Z M Z M Z M Z M Z × Z Z M × Z Z M Z M Z M Z Z Z Z × Z Z × Z Z Z Z Z M Z M Z M × Z Z M Z Z Z × Z Z Z M Z M Z M × Z Z M Z M Z M Z Z M × Z Z × Z Z M Z M Z Z Z × Z Z Z M Z M Z M Z × Z Z M × Z Z M Z M Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z × Z Z Z Z Z × Z Z Z M × Z Z M Z Z M Z M Z M Z M × Z Z M Z Z M Z M Z M Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z M × Z Z M Z Z M Z × Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z × Z Z Z Z M Z M Z M Z Z M Z M Z M Z M Z M Z Z Z M Z M Z M Z M Z M Z Z Z M Z M Z M Z M Z M Z Z Z M Z M Z M Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z M Z Z M Z Z M Z M Z M Z Z Z Z Z Z Z Z M Z M Z M Z Z M Z M Z M Z Z Z M Z M Z Z M Z Z M Z Z Z Z Z Z M Z M Z × Z Z M Z M Z M Z Z Z × Z Z Z Z Z M Z M Z × Z Z M Z M Z M Z Z Z × Z Z Z Z Z M Z M Z × Z Z M Z M Z M Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z Z Z × Z Z Z Z Z M Z M Z × Z Z M Z Z Z × Z Z Z M Z M Z × Z Z M Z Z Z × Z Z Z M Z M Z M Z M Z M Z M × Z Z M Z M Z M Z M Z M Z Z M Z M Z Z M × Z Z Z M Z M Z M Z M Z M Z M Z Z M Z × Z Z Z M Z M Z M Z Z Z Z Z Z × Z Z Z Z Z Z M Z M Z M Z M Z M Z M × Z Z Z M Z M Z M Z M Z Z M Z M Z Z M × Z Z Z M Z M Z M Z M Z M Z M Z M Z M Z × Z Z M Z M Z M Z M Z Z Z Z Z Z × Z Z Z Z Z Z M Z M Z M Z M Z Z M × Z Z M Z M Z M Z M Z M Z M Z M Z M Z M Z M × Z Z Z M Z M Z M Z Z Z Z Z Z × Z Z Z Z Z Z M Z M Z M Z Z M Z × Z Z Z M Z M Z M Z M Z Z M Z M Z Z M × Z Z Z M Z M Z M Z M Z M Z M Z M Z M Z M × Z Z Z M Z M Z M Z M Z M Z Z M Z Z × Z Z M Z M Z M Z M Z M Z M Z M Z Z M Z × Z Z Z M Z M Z M Z M Z M Z M Z M Z M Z M × Z Z M Z M Z M Z M Z M Z M Z M Z M × Z Z Z M Z M Z M Z Z M Z Z × Z Z M Z M Z Z Z Z Z Z × Z Z Z Z Z Z Z Z Z Z Z Z × Z Z Z × Z Z Z × Z Z Z × Z Z Z × Z Z Z × Z Z Z × Z Z Z × Z Z Z × Z Z Z M Z Z M Z M Z Z M Z Z Z Z Z Z Z M Z Z M Z Z Z Z M Z M Z × Z Z M Z Z Z × Z Z Z M Z M Z × Z Z M Z Z Z × Z Z Z M Z M Z × Z Z M Z Z Z × Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z M × Z Z M Z M × Z Z M Z Z M Z M × Z Z M Z Z M Z Z Z × Z Z Z Z Z × Z Z Z Z Z × Z Z Z Z Z × Z Z Z Z Z × Z Z Z Z Z × Z Z Z M Z M Z M Z Z M Z Z Z Z Z Z M Z M Z Z Z M Z M Z Z M Z Z M Z M Z M Z M × Z Z M Z M Z Z M × Z Z M Z M Z M Z M × Z Z M Z M Z Z M × Z Z M Z M Z M Z M Z M Z M Z M Z M × Z Z M Z M Z Z M Z M Z Z Z M × Z Z M Z M Z M Z M Z M Z M Z Z M × Z Z M Z M Z M Z M Z M Z Z M Z M × Z Z M Z Z Z Z M Z M Z Z Z M Z M Z Z Z M Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z M Z M Z M Z M Z M Z Z Z Z Z Z M Z M Z M Z M Z Z Z Z Z M Z Z M Z M Z M Z M Z Z M Z M Z M Z M Z M Z M Z Z M Z M Z M Z M Z Z M Z Z Z Z Z M Z M Z M Z Z Z Appendix O: Table: Invariants and SIs of TCIclassification generators for MSGs with non-trivialSI group1. Notations in the table of mapping The quantitative mappings between invariants and SIsfor all MSGs are tabulated in Table.XIV. For each MSGwith a nontrivial SI group, we list all the independent decorations as generators for TCI classification. How-ever, when the MSG has mirror symmetries, we give oneextra generator, that is, if the classification is Z m , we give m mirror decorations plus a Z decoration, and one canremove any one of the m mirror decorations in order toobtain m independent decorations. For each decoration,we show its invariants and SIs.For MSGs without SI, we also list their independentdecorations corresponding to the classification in Ta-7ble.XV.Here we explain the notations used in these two tables. a. The top row The top row gives the general in-formation of the MSG. The MSG number and MSG la-bel are given in Belov-Neronova-Smirnova (BNS) setting,followed by the TCI classification, where we separate themirror decorations and translations into Z M and Z , withthe subscript M denotes mirror. The SI group is givensubsequently as X BS , followed by the corresponding gen-erating SI name. b. Decorations The first column “deco” denotes thetype of decoration. As Z (translation) and Z decora-tions are generically not constructed by LCs, we just use“ Z ” and “ Z ” to denote them, respectively, while themirror decorations are all constructed by layers, whichwe use “ M ( mnl ; d )” to denote the mirror decorations,with ( mnl ; d ) representing Miller indices with respect tothe conventional lattice. The second column, “ Z n ,n , ··· ”gives the SI values for each generator. The third col-umn shows the name of MSG symmetries and their cor-responding invariants of the generators. c. Invariants The first set “weak=( w w w )” givesthe weak invariants along the three lattice bases of theprimitive cell, followed by other invariants defined inreal-space, including those of inversion, S , rotation,screw, glide, and their combinations with TRS. Fortype-4 MSGs, the invariant of half magnetic translation, { E | a } · T is given after the weak invariants. Note themirror invariants are given not by the real-space ones,but the momentum space mirror Chern numbers, whichappear at the end of the row for MSGs with mirror sym-metries. d. MSG elements We represent each MSG symme-try operation in the following short symbols: • rotation: n hkl , where n is the rotation angle and( hkl ) are the Miller indices of the rotation axis inthe conventional lattice. • screw: n hklt t t , where ( hkl ) denote the screw axisand ( t t t ) are the indices for the screw vector,both given in the conventional lattice. • inversion: i . In some MSGs, inversion could carryfractional translations and becomes i t t t , where( t t t ) is given in the conventional lattice. • S n : ¯ n hkl , where ( hkl ) denote the rotoinversion axisin the conventional lattice. Similar to inversion, itcould become ¯ n hklt t t in some MSGs. • glide: g hklt t t , where ( hkl ) denote the glide planeand ( t t t ) are the indices for the glide vector,both given in the conventional lattice. • anti-unitary symmetries: (cid:98) O ∗ , where ∗ denote theTRS and (cid:98) O can be symmetry operations men-tioned above plus the half lattice translation, e.g., { | } . e. Mirror Chern number Mirror Chern numbers aredenoted as Cm hkl ( n ) . Here ( hkl ) denote the Miller indicesof the mirror plane and n denotes the highest (unitary) C n rotation along the direction of the mirror, which isset to be 0 when there is no such rotation.There are two kinds of C m due to different Bravaislattices, i.e., C m = ( C + k ⊥ =0 , C − k ⊥ =0 , C + k ⊥ = π , C − k ⊥ = π ) or C m = ( C + k ⊥ =0 , C − k ⊥ =0 ), where k ⊥ is in the normal di-rection of mirror. This depends on whether there aretwo mirror-symmetric planes in the BZ, i.e., k ⊥ = 0 and k ⊥ = π , or there is only one mirror-symmetric plane inBZ, i.e., k ⊥ = 0.8 TABLE XIV: MSGs with nontrivial SI group2.4 P ¯1 Classification = Z × Z X BS = Z , , , SI = z P, , z P, , z P, , z P deco Z , , , weak iZ Z Z Z P S ¯1 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ iZ P Z X BS = Z SI = z C deco Z weak 2 Z P a Z X BS = Z SI = z C deco Z weak { | }∗ Z P /m Classification = Z M × Z X BS = Z , , SI = z +2 m, , z − m, , z +2 m,π deco Z , , weak 2 i Cm Z 110 (000) 0 1 1¯100 Z 100 (010) 0 0 1001M(010; 0) 111 (000) 0 0 1¯11¯1M(010; ) 111 (000) 0 0 1¯1¯1110.46 P (cid:48) /m (cid:48) Classification = Z × Z X BS = Z , , , SI = z P, , z P, , z P, , z P deco Z , , , weak 2 ∗ iZ Z Z P a /m Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak { | }∗ i Cm Z 10 (000) 1 0 1 1¯100M(010; 0) 11 (000) 0 0 0 1¯11¯1M(010; ) 11 (000) 0 0 0 1¯1¯1110.48 P b /m Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | }∗ i Cm Z P C /m Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | 12 12 }∗ i Cm Z P /m Classification = Z M × Z X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Z Z ) 0 (000) 0 0 2¯20011.54 P (cid:48) /m (cid:48) Classification = Z × Z X BS = Z , , SI = z P, , z P, , z P deco Z , , weak 2 ∗ iZ 002 (000) 1 1 Z 100 (¯100) 0 0 Z 010 (00¯1) 0 011.55 P a /m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Z ) 0 (000) 0 0 0 2¯20011.56 P b /m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Cm Z ) 0 (000) 0 0 0 2¯20011.57 P C /m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i 12 12 ∗ Cm Z ) 0 (000) 0 0 0 2¯20012.58 C /m Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 i Cm Z 01 (000) 0 1 1¯1 Z 11 (110) 0 1 20M(010; 0) 00 (000) 0 0 2¯212.62 C (cid:48) /m (cid:48) Classification = Z × Z X BS = Z , , SI = z P, , z P, , z P deco Z , , weak 2 ∗ iZ 002 (000) 1 1 Z 100 (1¯10) 0 0 Z 010 (001) 0 012.63 C c /m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Z C a /m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i Cm Z P /c Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 iZ 01 (000) 0 1 Z 10 (010) 0 013.69 P (cid:48) /c (cid:48) Classification = Z × Z X BS = Z , , SI = z P, , z P, , z P deco Z , , weak 2 ∗ iZ 002 (000) 1 1 Z 002 (002) 0 1 Z 100 (100) 0 013.70 P a /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ iZ P b /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ iZ P c /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i ∗ Z P A /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i 12 12 ∗ Z P C /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ iZ P /c Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 iZ Z P (cid:48) /c (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z P deco Z , weak 2 12 12 ∗ i Z 02 (000) 1 1 Z 10 (100) 0 0 Z 02 (002) 0 114.80 P a /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 iZ P b /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Z P c /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 i ∗ Z P A /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ iZ P C /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i 12 12 ∗ Z C /c Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 iZ 01 (000) 0 1 Z 10 (¯1¯10) 0 015.89 C (cid:48) /c (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z P deco Z , weak 2 ∗ iZ 02 (000) 1 1 Z 02 (00¯2) 0 1 Z 12 (1¯10) 1 115.90 C c /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i ∗ Z C a /c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ iZ P (cid:48) (cid:48) Z × Z X BS = Z SI = z C deco Z weak 2 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 2 ∗ ∗ Z Z P (cid:48) (cid:48) Z × Z X BS = Z SI = z C deco Z weak 2 12 12 ∗ 12 12 ∗ Z Z C (cid:48) (cid:48) Z × Z X BS = Z SI = z C deco Z weak 2 ∗ ∗ Z Z P m (cid:48) m (cid:48) Z X BS = Z SI = z C deco Z weak 2 m ∗ m ∗ Z P c (cid:48) c (cid:48) Z X BS = Z SI = z (cid:48) C deco Z weak 2 g ∗ g ∗ Z P m (cid:48) a (cid:48) Z X BS = Z SI = z C deco Z weak 2 g ∗ g ∗ Z P b nc Z X BS = Z SI = z (cid:48) C,π deco Z weak { | }∗ ∗ g 12 12 g ∗ Z P b (cid:48) a (cid:48) Z X BS = Z SI = z C deco Z weak 2 g 12 12 ∗ g 12 12 ∗ Z P C nn Z X BS = Z SI = z (cid:48) C,π deco Z weak { | 12 12 }∗ g ∗ g ∗ Z Cm (cid:48) m (cid:48) Z X BS = Z SI = z C deco Z weak 2 m ∗ m ∗ Z Cc (cid:48) c (cid:48) Z X BS = Z SI = z (cid:48) C deco Z weak 2 g ∗ g ∗ Z C a cc Z X BS = Z SI = z , . deco Z weak { | 12 12 }∗ g g Z Ab (cid:48) m (cid:48) Z X BS = Z SI = z (cid:48) C deco Z weak 2 g ∗ g ∗ Z Ab (cid:48) a (cid:48) Z X BS = Z SI = z , . deco Z weak 2 g ∗ g ∗ Z F m (cid:48) m (cid:48) Z X BS = Z SI = z , . deco Z weak 2 m ∗ m ∗ Z Ib (cid:48) a (cid:48) Z X BS = Z SI = z (cid:48) C deco Z weak 2 g ∗ g ∗ Z P mmm Classification = Z M X BS = Z , , , SI = z (cid:48) P, , z (cid:48) P, , z (cid:48) P, , z (cid:48) P deco Z , , , weak 2 i Cm Cm Cm Z ) 1000 (000) 0 0 0 0 1¯1¯11 0000 0000M(010; 0) 0102 (000) 0 0 0 0 0000 1¯11¯1 0000M(010; ) 0100 (000) 0 0 0 0 0000 1¯1¯11 0000M(001; 0) 0012 (000) 0 0 0 0 0000 0000 1¯11¯1M(001; ) 0010 (000) 0 0 0 0 0000 0000 1¯1¯1147.252 P m (cid:48) m (cid:48) m Classification = Z M × Z X BS = Z , , SI = z +2 m, , z − m, , z +2 m,π deco Z , , weak 2 ∗ i ∗ Cm Z 110 (000) 0 1 1 1 1¯100 Z 100 (001) 0 0 0 0 1001M(001; 0) 111 (000) 0 0 0 0 1¯11¯1M(001; ) 111 (000) 0 0 0 0 1¯1¯1147.254 P a mmm Classification = Z M X BS = Z , , SI = z (cid:48) P, , z (cid:48) P, , z (cid:48) P deco Z , , weak { | }∗ i Cm Cm Cm Z 003 (000) 1 0 0 1 0 1¯100 1¯100 1¯100M(100; 0) 002 (000) 0 0 0 0 0 2¯200 0000 0000M(010; 0) 102 (000) 0 0 0 0 0 0000 1¯11¯1 0000 M(010; ) 100 (000) 0 0 0 0 0 0000 1¯1¯11 0000M(001; 0) 012 (000) 0 0 0 0 0 0000 0000 1¯11¯1M(001; ) 010 (000) 0 0 0 0 0 0000 0000 1¯1¯1147.255 P C mmm Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak { | 12 12 }∗ i Cm Cm Cm Z 03 (000) 1 0 0 1 0 1¯100 1¯100 1¯100M(010; 0) 02 (000) 0 0 0 0 0 2¯200 0000 0000M(001; 0) 12 (000) 0 0 0 0 0 0000 1¯11¯1 0000M(001; ) 10 (000) 0 0 0 0 0 0000 1¯1¯11 0000M(100; 0) 02 (000) 0 0 0 0 0 0000 0000 2¯20047.256 P I mmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm Cm Cm Z P nnn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 i 12 12 Z P n (cid:48) n (cid:48) n Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 ∗ i 12 12 ∗ Z Z P c nnn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ i ∗ Z P C nnn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 i 12 12 ∗ 12 12 Z P I nnn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 i 12 12 12 ∗ 12 12 Z P ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Z P c (cid:48) c (cid:48) m Classification = Z M × Z X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak 2 ∗ i ∗ Cm Z 10 (000) 0 1 1 1 1¯100 Z 01 (002) 0 0 1 0 2011M(001; 0) 00 (000) 0 0 0 0 2¯20049.270 P c (cid:48) cm (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 1 0 1 1 Z 10 (0¯10) 0 0 0 049.272 P a ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Z P c ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Cm Z P B ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i Cm Z P C ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i Cm Z P I ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm Z P ban Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 i Z P b (cid:48) a (cid:48) n Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 12 12 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 10 (00¯1) 0 0 0 050.282 P b (cid:48) an (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 ∗ i ∗ Z Z P a ban Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 i ∗ Z P c ban Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 i Z P A ban Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i 12 12 Z P C ban Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 i 12 12 ∗ Z P I ban Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 i 12 12 12 ∗ Z P mma Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak 2 i Cm Cm Z 10 (000) 0 0 1 0 1¯100 1¯100M(100; ) 00 (000) 0 0 0 0 2¯200 0000M(010; 0) 11 (000) 0 0 0 0 0000 1¯11¯1M(010; ) 11 (000) 0 0 0 0 0000 1¯1¯1151.294 P m (cid:48) m (cid:48) a Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 10 (00¯1) 0 0 0 051.295 P mm (cid:48) a (cid:48) Classification = Z M × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ ∗ i Cm Z Z ) 0 (000) 0 0 0 0 2¯20051.296 P m (cid:48) ma (cid:48) Classification = Z M × Z X BS = Z , , SI = z +2 m, , z − m, , z +2 m,π deco Z , , weak 2 ∗ i ∗ Cm Z 110 (000) 1 0 1 1 1¯100 Z 100 (010) 0 0 0 0 1001M(010; 0) 111 (000) 0 0 0 0 1¯11¯1M(010; ) 111 (000) 0 0 0 0 1¯1¯1151.298 P a mma Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak { | }∗ ∗ i ∗ ∗ Cm Cm Z 10 (000) 1 0 1 0 1 1¯100 1¯100M(100; ) 00 (000) 0 0 0 0 0 2¯200 0000M(010; 0) 11 (000) 0 0 0 0 0 0000 1¯11¯1M(010; ) 11 (000) 0 0 0 0 0 0000 1¯1¯1151.299 P b mma Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | }∗ i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(010; 0) 0 (000) 0 0 0 0 0 0000 2¯20051.300 P c mma Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak { | }∗ i Cm Cm Z 10 (000) 1 0 0 1 0 1¯100 1¯100M(100; ) 00 (000) 0 0 0 0 0 2¯200 0000M(010; 0) 11 (000) 0 0 0 0 0 0000 1¯11¯1M(010; ) 11 (000) 0 0 0 0 0 0000 1¯1¯1151.301 P A mma Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | 12 12 }∗ i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(010; 0) 0 (000) 0 0 0 0 0 0000 2¯20051.302 P B mma Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak { | }∗ ∗ ∗ i ∗ Cm Cm Z 10 (000) 1 1 1 0 0 1¯100 1¯100M(100; ) 00 (000) 0 0 0 0 0 2¯200 0000M(010; 0) 11 (000) 0 0 0 0 0 0000 1¯11¯1M(010; ) 11 (000) 0 0 0 0 0 0000 1¯1¯1151.303 P C mma Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | 12 12 }∗ 12 12 ∗ i 12 12 ∗ ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 0000 2¯20051.304 P I mma Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | 12 12 12 }∗ i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(010; 0) 0 (000) 0 0 0 0 0 0000 2¯20052.305 P nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 i 12 12 Z P n (cid:48) n (cid:48) a Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 ∗ i 12 12 ∗ Z Z P nn (cid:48) a (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ 12 12 12 ∗ i 12 12 Z Z P n (cid:48) na (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ 12 12 12 i 12 12 ∗ Z Z P a nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ 12 12 i ∗ Z P b nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ 12 12 ∗ i ∗ 12 12 Z P c nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i 12 12 ∗ Z P A nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ ∗ i Z P B nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 12 12 12 i ∗ Z P C nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ ∗ i 12 12 Z P I nna Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i ∗ Z P mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Z P m (cid:48) n (cid:48) a Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ i ∗ Z Z P mn (cid:48) a (cid:48) Classification = Z M × Z X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak 2 ∗ ∗ i Cm Z 10 (000) 1 1 1 0 1¯100 Z 01 (200) 0 0 1 0 2011M(100; 0) 00 (000) 0 0 0 0 2¯20053.328 P m (cid:48) na (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 1 0 1 1 Z 10 (0¯10) 0 0 0 053.330 P a mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Cm Z P b mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Z P c mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Cm Z P A mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 ∗ i 12 12 ∗ Cm Z P B mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i Cm Z P C mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 ∗ i Cm Z P I mna Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ ∗ i Cm Z P cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 i Z P c (cid:48) c (cid:48) a Classification = Z × Z X BS = Z , SI = z (cid:48) C , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 11 (002) 0 0 1 054.343 P cc (cid:48) a (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ ∗ i Z Z P c (cid:48) ca (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 1 0 1 1 Z 10 (0¯10) 0 0 0 054.346 P a cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ ∗ Z P b cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Z P c cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Z P A cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 12 ∗ i 12 12 ∗ Z P B cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i ∗ Z P C cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ 12 12 ∗ i 12 12 ∗ Z P I cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i ∗ Z P bam Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak 2 12 12 i 12 12 Cm Z 10 (000) 0 0 1 0 1¯100M(001; 0) 11 (000) 0 0 0 0 1¯11¯1M(001; ) 11 (000) 0 0 0 0 1¯1¯1155.357 P b (cid:48) a (cid:48) m Classification = Z M × Z X BS = Z , , SI = z +2 m, , z − m, , z +2 m,π deco Z , , weak 2 12 12 ∗ i 12 12 ∗ Cm Z 110 (000) 0 1 1 1 1¯100 Z 100 (001) 0 0 0 0 1001M(001; 0) 111 (000) 0 0 0 0 1¯11¯1M(001; ) 111 (000) 0 0 0 0 1¯1¯1155.358 P b (cid:48) am (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ 12 12 i 12 12 ∗ Z Z P a bam Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak { | }∗ ∗ ∗ i ∗ 12 12 Cm Z 10 (000) 1 1 1 0 0 1¯100M(001; 0) 11 (000) 0 0 0 0 0 1¯11¯1M(001; ) 11 (000) 0 0 0 0 0 1¯1¯1155.361 P c bam Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | }∗ 12 12 i 12 12 Cm Z P A bam Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | 12 12 }∗ 12 12 12 12 i Cm Z P C bam Classification = Z M X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak { | 12 12 }∗ ∗ i ∗ Cm Z 10 (000) 1 0 1 1 1 1¯100M(001; 0) 11 (000) 0 0 0 0 0 1¯11¯1M(001; ) 11 (000) 0 0 0 0 0 1¯1¯1155.364 P I bam Classification = Z M X BS = Z SI = z +2 m, deco Z weak { | 12 12 12 }∗ ∗ i ∗ Cm Z P ccn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 12 i Z P c (cid:48) c (cid:48) n Classification = Z × Z X BS = Z , SI = z (cid:48) C , z (cid:48) P deco Z , weak 2 12 12 12 12 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 11 (002) 0 0 1 056.370 P c (cid:48) cn (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 ∗ 12 12 i ∗ Z Z P b ccn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ i ∗ 12 12 12 ∗ Z P c ccn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ i ∗ Z P A ccn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ Z P C ccn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ i 12 12 ∗ 12 12 ∗ Z P I ccn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 ∗ i 12 12 12 ∗ ∗ Z P bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 i Cm Z ) 0 (000) 0 0 0 0 2¯20057.382 P b (cid:48) c (cid:48) m Classification = Z M × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 ∗ i ∗ Cm Z Z ) 0 (000) 0 0 0 0 2¯20057.383 P bc (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ 12 12 ∗ i Z 01 (000) 1 1 1 0 Z 10 (¯100) 0 0 0 057.384 P b (cid:48) cm (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ 12 12 i ∗ Z Z P a bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 i Cm Z ) 0 (000) 0 0 0 0 0 2¯20057.387 P b bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ 12 12 ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20057.388 P c bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ ∗ i ∗ 12 12 Cm Z ) 0 (000) 0 0 0 0 0 2¯20057.389 P A bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ 12 12 i 12 12 ∗ ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20057.390 P B bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 12 12 12 ∗ i ∗ ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20057.391 P C bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 12 ∗ ∗ i 12 12 ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20057.392 P I bcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 ∗ ∗ i 12 12 12 ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯200 P nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 i 12 12 12 Cm Z P n (cid:48) n (cid:48) m Classification = Z M × Z X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak 2 12 12 12 ∗ i 12 12 12 ∗ Cm Z 10 (000) 0 1 1 1 1¯100 Z 01 (002) 0 0 1 0 2011M(001; 0) 00 (000) 0 0 0 0 2¯20058.398 P nn (cid:48) m (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ 12 12 12 ∗ i 12 12 12 Z Z P a nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 12 12 12 ∗ i ∗ ∗ Cm Z P c nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ i 12 12 ∗ Cm Z P B nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i Cm Z P C nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ Cm Z P I nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i ∗ Cm Z P mmn Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 i Cm Cm Z ) 0 (000) 0 0 0 0 2¯200 0000M(010; ) 0 (000) 0 0 0 0 0000 2¯20059.409 P m (cid:48) m (cid:48) n Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 12 12 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 10 (00¯1) 0 0 0 059.410 P mm (cid:48) n (cid:48) Classification = Z M × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 ∗ ∗ i Cm Z Z ) 0 (000) 0 0 0 0 2¯20059.412 P b mmn Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ i ∗ 12 12 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(010; ) 0 (000) 0 0 0 0 0 0000 2¯20059.413 P c mmn Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(010; ) 0 (000) 0 0 0 0 0 0000 2¯20059.414 P B mmn Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 12 ∗ 12 12 i ∗ ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(100; ) 0 (000) 0 0 0 0 0 0000 2¯20059.415 P C mmn Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ i 12 12 ∗ ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(100; ) 0 (000) 0 0 0 0 0 0000 2¯20059.416 P I mmn Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ 12 12 ∗ i 12 12 12 ∗ 12 12 Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 0000M(100; ) 0 (000) 0 0 0 0 0 0000 2¯20060.417 P bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 i 12 12 Z P b (cid:48) c (cid:48) n Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 ∗ i 12 12 ∗ Z Z P bc (cid:48) n (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 ∗ ∗ i 12 12 Z Z P b (cid:48) cn (cid:48) Classification = Z × Z X BS = Z , SI = z (cid:48) P , z , . deco Z , weak 2 12 12 12 ∗ i 12 12 ∗ Z 11 (000) 1 0 1 1 Z 10 (0¯20) 0 0 1 060.426 P a bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i ∗ 12 12 12 Z P b bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i Z P c bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 12 ∗ 12 12 ∗ i ∗ Z P A bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 12 12 12 i 12 12 ∗ Z P B bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ i ∗ ∗ Z P C bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ Z P I bcn Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i ∗ Z P bca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 i 12 12 Z P b (cid:48) c (cid:48) a Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 ∗ i 12 12 ∗ Z Z P a bca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 12 ∗ i ∗ ∗ Z P C bca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 i ∗ Z P I bca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ ∗ i 12 12 12 ∗ ∗ Z P nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i 12 12 12 Cm Z ) 0 (000) 0 0 0 0 2¯20062.446 P n (cid:48) m (cid:48) a Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ i 12 12 12 ∗ Z Z P nm (cid:48) a (cid:48) Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ ∗ i 12 12 12 Z Z P n (cid:48) ma (cid:48) Classification = Z M × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ i 12 12 12 ∗ Cm Z Z ) 0 (000) 0 0 0 0 2¯20062.450 P a nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ 12 12 ∗ i ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20062.451 P b nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ 12 12 12 ∗ i ∗ 12 12 12 Cm Z ) 0 (000) 0 0 0 0 0 2¯20062.452 P c nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ 12 12 12 i ∗ ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20062.453 P A nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 12 12 12 ∗ i 12 12 ∗ ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20062.454 P B nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Cm Z M(010; ) 0 (000) 0 0 0 0 0 2¯20062.455 P C nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 12 i 12 12 ∗ ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20062.456 P I nma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯20063.457 Cmcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Cm Z ) 0 (000) 0 0 0 0 2¯200 00M(100; 0) 0 (000) 0 0 0 0 0000 2¯263.462 Cm (cid:48) c (cid:48) m Classification = Z M × Z X BS = Z SI = z (cid:48) P deco Z weak 2 ∗ i ∗ Cm Z Z ) 0 (000) 0 0 0 0 2¯20063.463 Cmc (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i Cm Z 01 (000) 1 1 1 0 1¯1 Z 11 (¯110) 1 1 1 0 20M(100; 0) 00 (000) 0 0 0 0 2¯263.464 Cm (cid:48) cm (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 1 0 1 1 Z 10 (110) 0 0 0 063.466 C c mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 00M(100; 0) 0 (000) 0 0 0 0 0 0000 2¯263.467 C a mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 00M(100; 0) 0 (000) 0 0 0 0 0 0000 2¯263.468 C A mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ 12 12 ∗ i 12 12 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 00M(100; 0) 0 (000) 0 0 0 0 0 0000 2¯264.469 Cmca Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 12 i Cm Z Cm (cid:48) c (cid:48) a Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 12 12 12 12 ∗ i ∗ Z Z Cmc (cid:48) a (cid:48) Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 12 12 ∗ 12 12 ∗ i Cm Z 01 (000) 1 1 1 0 1¯1 Z 11 (1¯10) 1 1 1 0 20M(100; 0) 00 (000) 0 0 0 0 2¯264.476 Cm (cid:48) ca (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 12 12 ∗ 12 12 i ∗ Z 01 (000) 1 0 1 1 Z 10 (110) 1 1 0 064.478 C c mca Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i ∗ 12 12 Cm Z C a mca Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ ∗ i ∗ 12 12 Cm Z C A mca Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ ∗ i Cm Z ) 0 (000) 0 0 0 0 0 2¯265.481 Cmmm Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 2 i Cm Cm Cm Z 03 (000) 0 0 1 0 1¯1 1¯100 1¯1M(010; 0) 02 (000) 0 0 0 0 2¯2 0000 00M(001; 0) 12 (000) 0 0 0 0 00 1¯11¯1 00M(001; ) 10 (000) 0 0 0 0 00 1¯1¯11 00M(100; 0) 02 (000) 0 0 0 0 00 0000 2¯265.485 Cm (cid:48) m (cid:48) m Classification = Z M × Z X BS = Z , , SI = z +2 m, , z − m, , z +2 m,π deco Z , , weak 2 ∗ i ∗ Cm Z 110 (000) 0 1 1 1 1¯100 Z 100 (00¯1) 0 0 0 0 1001M(001; 0) 111 (000) 0 0 0 0 1¯11¯1M(001; ) 111 (000) 0 0 0 0 1¯1¯1165.486 Cmm (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i Cm Z 01 (000) 1 1 1 0 1¯1 Z 11 (¯110) 1 1 1 0 20M(100; 0) 00 (000) 0 0 0 0 2¯265.488 C c mmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Cm Cm Z C a mmm Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak { | 12 12 }∗ i Cm Cm Cm Z 03 (000) 1 0 0 1 0 1¯1 1¯1 1¯100M(100; 0) 02 (000) 0 0 0 0 0 2¯2 00 0000M(010; 0) 02 (000) 0 0 0 0 0 00 2¯2 0000M(001; 0) 12 (000) 0 0 0 0 0 00 00 1¯11¯1M(001; ) 10 (000) 0 0 0 0 0 00 00 1¯1¯1165.490 C A mmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm Cm Cm Z ) 2 (000) 0 0 0 0 0 0000 00 2¯266.491 Cccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Z Cc (cid:48) c (cid:48) m Classification = Z M × Z X BS = Z , SI = z +2 m, , z +2 m,π deco Z , weak 2 ∗ i ∗ Cm Z 10 (000) 0 1 1 1 1¯100 Z 01 (00¯2) 0 0 1 0 2011M(001; 0) 00 (000) 0 0 0 0 2¯20066.496 Ccc (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i Z 01 (000) 1 1 1 0 Z 11 (¯110) 1 1 1 066.498 C c ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Cm Z C a ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i Cm Z C A ccm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ ∗ i ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 0000 2¯267.501 Cmma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Cm Z ) 0 (000) 0 0 0 0 2¯2 00M(100; 0) 0 (000) 0 0 0 0 00 2¯267.505 Cm (cid:48) m (cid:48) a Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 10 (001) 0 0 0 067.506 Cmm (cid:48) a (cid:48) Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i Cm Z 01 (000) 1 1 1 0 1¯1 Z 11 (1¯10) 1 1 1 0 20M(100; 0) 00 (000) 0 0 0 0 2¯267.508 C c mma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯2 00M(100; 0) 0 (000) 0 0 0 0 0 00 2¯267.509 C a mma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯2 00M(100; 0) 0 (000) 0 0 0 0 0 00 2¯267.510 C A mma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ 12 12 ∗ i 12 12 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 00M(100; 0) 0 (000) 0 0 0 0 0 0000 2¯2 Ccca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 i 12 12 Z Cc (cid:48) c (cid:48) a Classification = Z × Z X BS = Z , SI = z (cid:48) P , z , . deco Z , weak 2 ∗ i 12 12 ∗ Z 10 (000) 0 1 1 1 Z 11 (002) 0 0 1 068.516 Ccc (cid:48) a (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i 12 12 Z 01 (000) 1 1 1 0 Z 11 (1¯10) 1 1 1 068.518 C c cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ Z C a cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ i ∗ ∗ Z C A cca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i 12 12 ∗ 12 12 Z F mmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Cm Cm Z ) 2 (000) 0 0 0 0 00 00 2¯269.524 F m (cid:48) m (cid:48) m Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Cm Z 01 (000) 0 1 1 1 1¯1 Z 11 (¯1¯10) 0 1 1 1 20M(001; 0) 00 (000) 0 0 0 0 2¯269.526 F S mmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm Cm Cm Z F ddd Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 14 14 i 14 14 Z F d (cid:48) d (cid:48) d Classification = Z × Z X BS = Z SI = z (cid:48) P deco Z weak 2 14 14 ∗ i 14 14 ∗ Z Z F S ddd Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 14 14 i 12 12 12 ∗ 14 14 Z Immm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Cm Cm Z Im (cid:48) m (cid:48) m Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Cm Z 01 (000) 0 1 1 1 1¯1 Z 11 (11¯1) 0 1 1 1 20M(001; 0) 00 (000) 0 0 0 0 2¯271.538 I c mmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i Cm Cm Cm Z Ibam Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Z Ib (cid:48) a (cid:48) m Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Cm Z 01 (000) 0 1 1 1 1¯1 Z 11 (¯1¯11) 0 0 1 0 20M(001; 0) 00 (000) 0 0 0 0 2¯272.544 Iba (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i Z 01 (000) 1 1 1 0 Z 11 (1¯1¯1) 1 1 1 072.546 I c bam Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ Cm Z I b bam Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i Cm Z Ibca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 2 i Z Ib (cid:48) c (cid:48) a Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 10 (¯1¯11) 0 0 0 173.553 I c bca Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ Z Imma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 2 i Cm Cm Z ) 0 (000) 0 0 0 0 2¯2 00M(100; 0) 0 (000) 0 0 0 0 00 2¯274.558 Im (cid:48) m (cid:48) a Classification = Z × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ i ∗ Z 01 (000) 0 1 1 1 Z 10 (11¯1) 0 0 0 074.559 Imm (cid:48) a (cid:48) Classification = Z M × Z X BS = Z , SI = z P, , z (cid:48) P deco Z , weak 2 ∗ ∗ i Cm Z 01 (000) 1 1 1 0 1¯1 Z 11 (1¯1¯1) 1 1 1 0 20M(100; 0) 00 (000) 0 0 0 0 2¯274.561 I c mma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i ∗ ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯2 00M(100; 0) 0 (000) 0 0 0 0 0 00 2¯274.562 I b mma Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 00 2¯275.1 P Z X BS = Z SI = z C deco Z weak 4 Z P C Z X BS = Z SI = z C / Z weak { | 12 12 }∗ Z P Classification = Z X BS = Z SI = z C / Z weak 4 Z P C Classification = Z X BS = Z SI = z C / Z weak { | 12 12 }∗ Z I Z X BS = Z SI = z C / Z weak 4 Z P ¯4 Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C deco Z , , weak ¯4 ¯4 Z 010 (000) 1 1 Z 001 (001) 0 081.36 P c ¯4 Classification = Z X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 ¯4 Z P C ¯4 Classification = Z X BS = Z , SI = z ,S , z C / Z , weak { | 12 12 }∗ ¯4 ¯4 Z 10 (000) 1 1 181.38 P I ¯4 Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 ¯4 Z I ¯4 Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 ¯4 Z 010 (000) 1 1 Z 001 (11¯1) 0 082.42 I c ¯4 Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 ¯4 Z P /m Classification = Z M × Z X BS = Z , , SI = z +4 m, , z − m, , z +4 m,π deco Z , , weak 4 i ¯4 ¯4 Cm Z 130 (000) 0 1 1 1 1¯100 Z 100 (001) 0 0 0 0 1001M(001; 0) 131 (000) 0 0 0 0 1¯11¯1M(001; ) 133 (000) 0 0 0 0 1¯1¯1183.45 P (cid:48) /m Classification = Z M X BS = Z , SI = z +2 m, , z (cid:48) P deco Z , weak 4 ∗ i ¯4 ∗ ¯4 ∗ Cm Z 03 (000) 1 1 0 0 1¯100M(001; 0) 12 (000) 0 0 0 0 1¯11¯1M(001; ) 10 (000) 0 0 0 0 1¯1¯11 P (cid:48) /m (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ¯4 ¯4 Z P c /m Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | }∗ i ¯4 ¯4 Cm Z P C /m Classification = Z M X BS = Z , SI = z +4 m, , z +4 m,π deco Z , weak { | 12 12 }∗ i ¯4 ¯4 Cm Z 10 (000) 1 0 1 1 1 1¯100M(001; 0) 11 (000) 0 0 0 0 0 1¯11¯1M(001; ) 13 (000) 0 0 0 0 0 1¯1¯1183.50 P I /m Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 12 }∗ i ¯4 ¯4 Cm Z P /m Classification = Z M × Z X BS = Z , SI = z +2 m,π , z +4 m, deco Z , weak 4 i ¯4 ¯4 Cm Z 01 (000) 0 1 1 1 1¯100 Z 12 (00¯2) ¯1 1 0 0 2011M(001; 0) 02 (000) 0 0 0 0 2¯20084.53 P (cid:48) /m Classification = Z M X BS = Z SI = z +2 m, deco Z weak 4 ∗ i ¯4 ∗ ¯4 ∗ Cm Z P (cid:48) /m (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ¯4 ¯4 Z P c /m Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | }∗ ∗ i ¯4 ∗ ¯4 ∗ Cm Z P C /m Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 }∗ i ¯4 ¯4 Cm Z P I /m Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 12 }∗ 12 12 ∗ i ¯4 12 12 ∗ ¯4 12 12 ∗ Cm Z P /n Classification = Z × Z X BS = Z , SI = z ,S , z C deco Z , weak 4 i ¯4 ¯4 Z 10 (000) 0 1 1 1 Z 01 (001) 0 0 0 085.61 P (cid:48) /n Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ¯4 ∗ ¯4 ∗ Z P (cid:48) /n (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ¯4 ¯4 Z P c /n Classification = Z X BS = Z SI = z ,S deco Z weak { | }∗ i ¯4 ¯4 Z P C /n Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ i 12 12 ∗ ¯4 ∗ ¯4 ∗ Z P I /n Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ i 12 12 12 ∗ ¯4 12 12 ∗ ¯4 ∗ Z P /n Classification = Z × Z X BS = Z , SI = z ,S , z C / Z , weak 4 12 12 i ¯4 12 12 ¯4 Z 10 (000) 0 1 1 1 Z 01 (00¯2) ¯1 1 0 086.69 P (cid:48) /n Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 12 12 ∗ i ¯4 12 12 ∗ ¯4 ∗ Z P (cid:48) /n (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 12 12 ∗ i ∗ ¯4 12 12 ¯4 Z P c /n Classification = Z X BS = Z SI = z ,S deco Z weak { | }∗ ∗ i ¯4 ∗ ¯4 ∗ Z P C /n Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ 12 12 i 12 12 ∗ ¯4 ∗ ¯4 12 12 ∗ Z P I /n Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ∗ i 12 12 12 ∗ ¯4 12 12 ¯4 Z I /m Classification = Z M × Z X BS = Z , SI = z +4 m, , z − m, deco Z , weak 4 i ¯4 ¯4 Cm Z 01 (000) 0 1 1 1 1¯1 Z 11 (11¯1) 0 1 1 1 20M(001; 0) 02 (000) 0 0 0 0 2¯287.77 I (cid:48) /m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ¯4 ∗ ¯4 ∗ Cm Z I (cid:48) /m (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ¯4 ¯4 Z I c /m Classification = Z M X BS = Z SI = z − m, deco Z weak { | 12 12 }∗ i ¯4 ¯4 Cm Z I /a Classification = Z × Z X BS = Z , SI = z ,S , z (cid:48) P deco Z , weak 4 14 14 14 i ¯4 14 14 14 ¯4 14 14 14 Z 11 (000) 0 1 1 1 Z 01 (22¯2) 1 1 0 088.83 I (cid:48) /a Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 14 14 14 ∗ i ¯4 14 14 14 ∗ ¯4 14 14 14 ∗ Z I (cid:48) /a (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 14 14 14 ∗ i ∗ ¯4 14 14 14 ¯4 14 14 14 Z I c /a Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ 14 14 14 i ¯4 14 14 14 ¯4 14 14 14 Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 4 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 4 12 12 12 12 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C / Z weak 4 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C / Z weak 4 12 12 12 12 12 12 ∗ ∗ Z Z I (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C / Z weak 4 ∗ ∗ Z Z P m (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 4 m ∗ m ∗ Z P b (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 4 g 12 12 ∗ g 12 12 ∗ Z P c (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C / Z weak 4 g ∗ m ∗ Z P n (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C / Z weak 4 12 12 12 g 12 12 12 ∗ m ∗ Z P c (cid:48) c (cid:48) Classification = Z X BS = Z SI = z (cid:48) C deco Z weak 4 g ∗ g ∗ Z P n (cid:48) c (cid:48) Classification = Z X BS = Z SI = z C / Z weak 4 g 12 12 12 ∗ g 12 12 12 ∗ Z P C nc Classification = Z X BS = Z SI = z , . deco Z weak { | 12 12 }∗ 12 12 ∗ g 12 12 12 g ∗ Z P m (cid:48) c (cid:48) Classification = Z X BS = Z SI = z C / Z weak 4 m ∗ g ∗ Z P b (cid:48) c (cid:48) Classification = Z X BS = Z SI = z C / Z weak 4 g 12 12 ∗ g 12 12 12 ∗ Z I m (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C / Z weak 4 m ∗ m ∗ Z I c (cid:48) m (cid:48) Classification = Z X BS = Z SI = z (cid:48) C deco Z weak 4 g ∗ g ∗ Z I c (cid:48) d (cid:48) Classification = Z X BS = Z SI = z , . deco Z weak 4 12 14 g ∗ g ∗ Z P ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak ¯4 Cm Z P ¯42 (cid:48) m (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C deco Z , , weak ¯4 ∗ m ∗ Z 010 (000) 1 1 0 Z 001 (001) 0 0 0111.256 P c ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 Cm Z P C ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 Cm Z P I ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 Cm Z P ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak ¯4 g Z P ¯42 (cid:48) c (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 ∗ g ∗ Z 010 (000) 1 1 0 Z 011 (00¯2) 1 0 ¯1112.264 P c ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 ∗ m ∗ Z P C ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 g Z P I ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 12 12 12 ∗ g 12 12 ∗ Z P ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak ¯4 12 12 Cm Z ) 0 (000) 0 0 2¯2113.271 P ¯42 (cid:48) m (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C deco Z , , weak ¯4 12 12 ∗ g 12 12 ∗ Z 010 (000) 1 1 0 Z 001 (001) 0 0 0113.272 P c ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 12 12 Cm Z ) 0 (000) 0 0 0 2¯2113.273 P C ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 12 12 ∗ ∗ Cm Z ) 0 (000) 0 0 0 2¯2113.274 P I ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 12 12 12 ∗ ∗ Cm Z ) 0 (000) 0 0 0 2¯2114.275 P ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak ¯4 12 12 12 g 12 12 12 Z P ¯42 (cid:48) c (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 12 12 12 ∗ g 12 12 12 ∗ Z 010 (000) 1 1 0 Z 011 (002) 1 0 1114.280 P c ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 ∗ 12 12 12 g 12 12 ∗ Z P C ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 12 12 ∗ g 12 12 12 ∗ Z P I ¯42 c Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 ∗ m ∗ Z P ¯4 m Z M X BS = Z SI = z ,S deco Z weak ¯4 Cm Z ) 0 (000) 0 0 1¯1¯11115.287 P ¯4 m (cid:48) (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C deco Z , , weak ¯4 m ∗ ∗ Z 010 (000) 1 0 1 Z 001 (001) 0 0 0115.288 P c ¯4 m Z M X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 Cm Z ) 0 (000) 0 0 0 1¯1¯11115.289 P C ¯4 m Z M X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 Cm Z P I ¯4 m Z M X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 Cm Z P ¯4 c Z X BS = Z SI = z ,S deco Z weak ¯4 g Z P ¯4 c (cid:48) (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 g ∗ ∗ Z 010 (000) 1 0 1 Z 011 (002) 1 1 0116.296 P c ¯4 c Z X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 ∗ m ∗ Z P C ¯4 c Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 g Z P I ¯4 c Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 g Z P ¯4 b Z X BS = Z SI = z ,S deco Z weak ¯4 g 12 12 12 12 Z P ¯4 b (cid:48) (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C deco Z , , weak ¯4 g 12 12 ∗ 12 12 ∗ Z 010 (000) 1 0 1 Z 001 (001) 0 0 0117.304 P c ¯4 b Z X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 g 12 12 12 12 Z P C ¯4 b Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 12 12 ∗ 12 12 m ∗ Z P I ¯4 b Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 g ∗ ∗ Z P ¯4 n Z X BS = Z SI = z ,S deco Z weak ¯4 g 12 12 12 12 12 12 Z P ¯4 n (cid:48) (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 g 12 12 12 ∗ 12 12 12 ∗ Z 010 (000) 1 0 1 Z 001 (002) 0 1 1118.312 P c ¯4 n Z X BS = Z SI = z ,S deco Z weak { | }∗ ¯4 ∗ g 12 12 ∗ 12 12 12 Z P C ¯4 n Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 12 12 ∗ 12 12 12 g ∗ Z P I ¯4 n Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 12 12 12 ∗ m ∗ 12 12 12 Z I ¯4 m Z M X BS = Z SI = z ,S deco Z weak ¯4 Cm Z I ¯4 m (cid:48) (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 m ∗ ∗ Z 010 (000) 1 0 1 Z 010 (11¯1) 1 0 1119.320 I c ¯4 m Z M X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 Cm Z I ¯4 c Z X BS = Z SI = z ,S deco Z weak ¯4 g Z I ¯4 c (cid:48) (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 g ∗ ∗ Z 010 (000) 1 0 1 Z 000 (¯1¯11) 0 ¯1 1120.326 I c ¯4 c Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 ∗ m ∗ Z I ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak ¯4 Cm Z I ¯42 (cid:48) m (cid:48) Classification = Z × Z X BS = Z , , SI = z , Weyl , z ,S , z C / Z , , weak ¯4 ∗ m ∗ Z 010 (000) 1 1 0 Z 010 (11¯1) 1 1 0121.332 I c ¯42 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 Cm Z I ¯42 d Classification = Z X BS = Z SI = z ,S deco Z weak ¯4 12 14 g 12 14 Z I ¯42 (cid:48) d (cid:48) Classification = Z × Z X BS = Z , SI = z , Weyl , z ,S deco Z , weak ¯4 12 14 ∗ g 12 14 ∗ Z 01 (000) 1 1 0 Z 00 (22¯2) 0 1 1122.338 I c ¯42 d Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 }∗ ¯4 g 12 14 12 14 Z P /mmm Classification = Z M X BS = Z , , SI = z (cid:48) P, , z +4 m,π , z deco Z , , weak 4 i ¯4 Cm Cm Cm Z 003 (000) 0 0 1 0 1 1¯100 1¯100 1¯1M(100; 0) 104 (000) 0 0 0 0 0 1¯11¯1 0000 00M(100; ) 100 (000) 0 0 0 0 0 1¯1¯11 0000 00M(001; 0) 016 (000) 0 0 0 0 0 0000 1¯11¯1 00M(001; ) 030 (000) 0 0 0 0 0 0000 1¯1¯11 00M(1¯10; 0) 004 (000) 0 0 0 0 0 0000 0000 2¯2123.342 P (cid:48) /mm (cid:48) m Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 4 ∗ ∗ i ¯4 ∗ Cm Cm Z 03 (000) 1 1 1 0 0 1¯100 1¯1M(001; 0) 12 (000) 0 0 0 0 0 1¯11¯1 00M(001; ) 10 (000) 0 0 0 0 0 1¯1¯11 00M(1¯10; 0) 00 (000) 0 0 0 0 0 0000 2¯2123.343 P (cid:48) /mmm (cid:48) Classification = Z M X BS = Z , , SI = z (cid:48) P, , z (cid:48) P, , z (cid:48) P deco Z , , weak 4 ∗ i ∗ ¯4 ∗ m ∗ Cm Cm Z 003 (000) 1 0 1 1 0 0 1¯100 1¯100M(100; 0) 100 (000) 0 0 0 0 0 0 1¯11¯1 0000M(100; ) 100 (000) 0 0 0 0 0 0 1¯1¯11 0000M(001; 0) 012 (000) 0 0 0 0 0 0 0000 1¯11¯1M(001; ) 010 (000) 0 0 0 0 0 0 0000 1¯1¯11123.344 P (cid:48) /m (cid:48) m (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 Cm Z P /mm (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , , SI = z +4 m, , z − m, , z +4 m,π deco Z , , weak 4 ∗ i ∗ ¯4 m ∗ Cm Z 130 (000) 0 1 1 1 1 0 1¯100 Z 100 (00¯1) 0 0 0 0 0 0 1001M(001; 0) 131 (000) 0 0 0 0 0 0 1¯11¯1M(001; ) 133 (000) 0 0 0 0 0 0 1¯1¯11123.346 P (cid:48) /m (cid:48) mm (cid:48) Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 m ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 1¯1¯11123.348 P c /mmm Classification = Z M X BS = Z , SI = z (cid:48) P, , z deco Z , weak { | }∗ i ¯4 Cm Cm Cm Z 03 (000) 1 0 0 1 0 1 1¯100 1¯100 1¯1M(100; 0) 14 (000) 0 0 0 0 0 0 1¯11¯1 0000 00M(100; ) 10 (000) 0 0 0 0 0 0 1¯1¯11 0000 00M(001; 0) 06 (000) 0 0 0 0 0 0 0000 2¯200 00M(1¯10; 0) 04 (000) 0 0 0 0 0 0 0000 0000 2¯2123.349 P C /mmm Classification = Z M X BS = Z , SI = z +4 m,π , z deco Z , weak { | 12 12 }∗ i ¯4 Cm Cm Cm Z 03 (000) 1 0 0 1 0 1 1¯1 1¯100 1¯100M(110; 0) 04 (000) 0 0 0 0 0 0 2¯2 0000 0000M(001; 0) 16 (000) 0 0 0 0 0 0 00 1¯11¯1 0000M(001; ) 10 (000) 0 0 0 0 0 0 00 1¯1¯11 0000M(100; 0) 04 (000) 0 0 0 0 0 0 00 0000 2¯200123.350 P I /mmm Classification = Z M X BS = Z SI = z deco Z weak { | 12 12 12 }∗ i ¯4 Cm Cm Cm Z P /mcc Classification = Z M X BS = Z SI = z +4 m, deco Z weak 4 i ¯4 g Cm Z ) 2 (000) 0 0 0 0 0 0 2¯200124.354 P (cid:48) /mc (cid:48) c Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ g Cm Z ) 2 (000) 0 0 0 0 0 0 2¯200124.355 P (cid:48) /mcc (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ∗ ¯4 ∗ g ∗ Cm Z ) 2 (000) 0 0 0 0 0 0 2¯200124.356 P (cid:48) /m (cid:48) c (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 g Z P /mc (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +4 m, , z +4 m,π deco Z , weak 4 ∗ i ∗ ¯4 g ∗ Cm Z 10 (000) 0 1 1 1 1 0 1¯100 Z 21 (00¯2) 0 0 1 0 1 ¯1 2011M(001; ) 20 (000) 0 0 0 0 0 0 2¯200124.358 P (cid:48) /m (cid:48) cc (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 g ∗ Z P c /mcc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | }∗ ∗ i ∗ ¯4 m ∗ Cm Z P C /mcc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 }∗ i ¯4 g Cm Z ) 2 (000) 0 0 0 0 0 0 0 2¯200124.362 P I /mcc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 12 }∗ i ¯4 g Cm Z P /nbm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 i ¯4 Cm Z P (cid:48) /nb (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ Cm Z P (cid:48) /nbm (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ∗ ¯4 ∗ g 12 12 ∗ Z P (cid:48) /n (cid:48) b (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 Cm Z P /nb (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C deco Z , weak 4 ∗ i ∗ ¯4 g 12 12 ∗ Z 10 (000) 0 1 1 1 1 0 Z 01 (001) 0 0 0 0 0 0125.370 P (cid:48) /n (cid:48) bm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 g 12 12 ∗ Z P c /nbm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i ¯4 Cm Z P C /nbm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ 12 12 ∗ i 12 12 ∗ ¯4 g ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 0 2¯2125.374 P I /nbm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i 12 12 12 ∗ ¯4 12 12 ∗ Cm Z P /nnc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 12 12 i ¯4 g 12 12 12 Z P (cid:48) /nn (cid:48) c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 ∗ i ¯4 ∗ g 12 12 12 Z P (cid:48) /nnc (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 i ∗ ¯4 ∗ g 12 12 12 ∗ Z P (cid:48) /n (cid:48) n (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 i ∗ ∗ ¯4 g 12 12 12 Z P /nn (cid:48) c (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C / Z , weak 4 12 12 ∗ i ∗ ¯4 g 12 12 12 ∗ Z 10 (000) 0 1 1 1 1 0 Z 01 (00¯2) 0 1 0 1 0 ¯1126.382 P (cid:48) /n (cid:48) nc (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 ∗ i ∗ ¯4 g 12 12 12 ∗ Z P c /nnc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ ¯4 g 12 12 ∗ Z P C /nnc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ 12 12 12 ∗ i 12 12 ∗ ¯4 g ∗ Z P I /nnc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 i 12 12 12 ∗ ¯4 12 12 ∗ m ∗ Z P /mbm Classification = Z M X BS = Z , SI = z +4 m, , z +4 m,π deco Z , weak 4 12 12 i 12 12 ¯4 Cm Cm Z 10 (000) 0 0 1 0 1 1¯1 1¯100M(1¯10; ) 00 (000) 0 0 0 0 0 2¯2 0000M(001; 0) 11 (000) 0 0 0 0 0 00 1¯11¯1M(001; ) 13 (000) 0 0 0 0 0 00 1¯1¯11127.390 P (cid:48) /mb (cid:48) m Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 4 ∗ 12 12 ∗ i 12 12 ¯4 ∗ Cm Cm Z 03 (000) 1 1 1 0 0 1¯1 1¯100M(1¯10; ) 00 (000) 0 0 0 0 0 2¯2 0000M(001; 0) 12 (000) 0 0 0 0 0 00 1¯11¯1M(001; ) 10 (000) 0 0 0 0 0 00 1¯1¯11127.391 P (cid:48) /mbm (cid:48) Classification = Z M X BS = Z , SI = z +2 m,π , z (cid:48) P deco Z , weak 4 ∗ 12 12 i 12 12 ∗ ¯4 ∗ g 12 12 ∗ Cm Z 03 (000) 1 0 1 1 0 0 1¯100M(001; 0) 12 (000) 0 0 0 0 0 0 1¯11¯1M(001; ) 10 (000) 0 0 0 0 0 0 1¯1¯11127.392 P (cid:48) /m (cid:48) b (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 i ∗ 12 12 ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2127.393 P /mb (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , , SI = z +4 m, , z − m, , z +4 m,π deco Z , , weak 4 12 12 ∗ i 12 12 ∗ ¯4 g 12 12 ∗ Cm Z 130 (000) 0 1 1 1 1 0 1¯100 Z 100 (00¯1) 0 0 0 0 0 0 1001M(001; 0) 131 (000) 0 0 0 0 0 0 1¯11¯1M(001; ) 133 (000) 0 0 0 0 0 0 1¯1¯11127.394 P (cid:48) /m (cid:48) bm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 ∗ i ∗ 12 12 ¯4 g 12 12 ∗ Z P c /mbm Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | }∗ 12 12 i 12 12 ¯4 Cm Cm Z M(1¯10; ) 0 (000) 0 0 0 0 0 0 2¯2 0000M(001; 0) 2 (000) 0 0 0 0 0 0 00 2¯200127.397 P C /mbm Classification = Z M X BS = Z , SI = z +4 m, , z +4 m,π deco Z , weak { | 12 12 }∗ 12 12 ∗ 12 12 i ∗ ¯4 12 12 ∗ m ∗ Cm Cm Z 10 (000) 1 1 0 1 1 0 0 1¯1 1¯100M(110; ) 00 (000) 0 0 0 0 0 0 0 2¯2 0000M(001; 0) 11 (000) 0 0 0 0 0 0 0 00 1¯11¯1M(001; ) 13 (000) 0 0 0 0 0 0 0 00 1¯1¯11127.398 P I /mbm Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 12 }∗ ∗ i ∗ ¯4 Cm Cm Z ) 0 (000) 0 0 0 0 0 0 0000 2¯2128.399 P /mnc Classification = Z M X BS = Z SI = z +4 m, deco Z weak 4 12 12 12 i 12 12 12 ¯4 g 12 12 12 Cm Z P (cid:48) /mn (cid:48) c Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 12 ∗ i 12 12 12 ¯4 ∗ g 12 12 12 Cm Z P (cid:48) /mnc (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 12 i 12 12 12 ∗ ¯4 ∗ g 12 12 12 ∗ Cm Z P (cid:48) /m (cid:48) n (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 12 i ∗ 12 12 12 ∗ ¯4 g 12 12 12 Z P /mn (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +4 m, , z +4 m,π deco Z , weak 4 12 12 12 ∗ i 12 12 12 ∗ ¯4 g 12 12 12 ∗ Cm Z 10 (000) 0 1 1 1 1 0 1¯100 Z 21 (002) 0 0 1 0 1 1 2011M(001; 0) 20 (000) 0 0 0 0 0 0 2¯200128.406 P (cid:48) /m (cid:48) nc (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 12 ∗ i ∗ 12 12 12 ¯4 g 12 12 12 ∗ Z P c /mnc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | }∗ 12 12 ∗ i 12 12 ∗ ¯4 g 12 12 ∗ Cm Z P C /mnc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 12 i ∗ ¯4 12 12 ∗ g ∗ Cm Z ) 2 (000) 0 0 0 0 0 0 0 2¯200128.410 P I /mnc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 12 }∗ ∗ i ∗ ¯4 m ∗ Cm Z P /nmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 i 12 12 ¯4 Cm Cm Z ) 0 (000) 0 0 0 0 0 2¯200 00M(1¯10; 0) 0 (000) 0 0 0 0 0 0000 2¯2 P (cid:48) /nm (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i 12 12 ¯4 ∗ Cm Z P (cid:48) /nmm (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i 12 12 ∗ ¯4 ∗ m ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200129.416 P (cid:48) /n (cid:48) m (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ 12 12 ∗ ¯4 Cm Z P /nm (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C deco Z , weak 4 ∗ i 12 12 ∗ ¯4 m ∗ Z 10 (000) 0 1 1 1 1 0 Z 03 (00¯1) 0 0 0 0 0 0129.418 P (cid:48) /n (cid:48) mm (cid:48) Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ 12 12 ¯4 m ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200129.420 P c /nmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i 12 12 ¯4 Cm Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200 00M(1¯10; 0) 0 (000) 0 0 0 0 0 0 0000 2¯2129.421 P C /nmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i 12 12 ∗ ∗ ¯4 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2 0000M(100; ) 0 (000) 0 0 0 0 0 0 00 2¯200129.422 P I /nmm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 ∗ i 12 12 12 ∗ ∗ ¯4 12 12 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200 00M(1¯10; 0) 0 (000) 0 0 0 0 0 0 0000 2¯2130.423 P /ncc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 i 12 12 12 ¯4 g Z P (cid:48) /nc (cid:48) c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i 12 12 12 ¯4 ∗ g Z P (cid:48) /ncc (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i 12 12 12 ∗ ¯4 ∗ g ∗ Z P (cid:48) /n (cid:48) c (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ 12 12 12 ∗ ¯4 g Z P /nc (cid:48) c (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z (cid:48) C deco Z , weak 4 ∗ i 12 12 12 ∗ ¯4 g ∗ Z 10 (000) 0 1 1 1 1 0 Z 11 (002) 0 0 1 0 1 1130.430 P (cid:48) /n (cid:48) cc (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ 12 12 12 ¯4 g ∗ Z P c /ncc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i 12 12 ∗ ¯4 m ∗ Z P C /ncc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i 12 12 ∗ ∗ ¯4 ∗ g Z P I /ncc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i 12 12 12 ∗ ∗ ¯4 12 12 ∗ g Z P /mmc Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 4 i ¯4 g Cm Cm Z 03 (000) 0 0 1 0 1 1 1¯100 1¯100M(100; 0) 10 (000) 0 0 0 0 0 0 1¯11¯1 0000M(100; ) 10 (000) 0 0 0 0 0 0 1¯1¯11 0000M(001; ) 02 (000) 0 0 0 0 0 0 0000 2¯200131.438 P (cid:48) /mm (cid:48) c Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ g Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200131.439 P (cid:48) /mmc (cid:48) Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 4 ∗ i ∗ ¯4 ∗ g ∗ Cm Cm Z 03 (000) 1 0 1 1 0 0 1¯100 1¯100M(100; 0) 10 (000) 0 0 0 0 0 0 1¯11¯1 0000M(100; ) 10 (000) 0 0 0 0 0 0 1¯1¯11 0000M(001; ) 02 (000) 0 0 0 0 0 0 0000 2¯200131.440 P (cid:48) /m (cid:48) m (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 g Z P /mm (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +2 m,π , z +4 m, deco Z , weak 4 ∗ i ∗ ¯4 g ∗ Cm Z 01 (000) 0 1 1 1 1 0 1¯100 Z 12 (00¯2) ¯1 1 1 0 0 ¯1 2011M(001; ) 02 (000) 0 0 0 0 0 0 2¯200131.442 P (cid:48) /m (cid:48) mc (cid:48) Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 g ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 1¯1¯11131.444 P c /mmc Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak { | }∗ ∗ i ∗ ¯4 ∗ m ∗ Cm Cm Z 03 (000) 1 1 0 1 1 0 0 1¯100 1¯100M(100; 0) 10 (000) 0 0 0 0 0 0 0 1¯11¯1 0000M(100; ) 10 (000) 0 0 0 0 0 0 0 1¯1¯11 0000M(001; 0) 02 (000) 0 0 0 0 0 0 0 0000 2¯200131.445 P C /mmc Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i ¯4 Cm Cm Z ) 2 (000) 0 0 0 0 0 0 0000 2¯200131.446 P I /mmc Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 ∗ i 12 12 ∗ ¯4 12 12 ∗ g 12 12 ∗ Cm Cm Z M(001; 0) 2 (000) 0 0 0 0 0 0 0 0000 2¯200132.447 P /mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 i ¯4 Cm Cm Z ) 2 (000) 0 0 0 0 0 00 2¯200132.450 P (cid:48) /mc (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ Cm Cm Z ) 2 (000) 0 0 0 0 0 00 2¯200132.451 P (cid:48) /mcm (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ∗ ¯4 ∗ m ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200132.452 P (cid:48) /m (cid:48) c (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 Cm Z P /mc (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z +2 m,π , z +4 m, deco Z , weak 4 ∗ i ∗ ¯4 m ∗ Cm Z 01 (000) 0 1 1 1 1 0 1¯100 Z 12 (002) 1 0 1 1 0 0 2011M(001; ) 02 (000) 0 0 0 0 0 0 2¯200132.454 P (cid:48) /m (cid:48) cm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 m ∗ Z P c /mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i ¯4 ∗ Cm Cm Z P C /mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ i ¯4 g Cm Cm Z ) 2 (000) 0 0 0 0 0 0 0 00 2¯200132.458 P I /mcm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 ∗ i 12 12 12 ∗ ¯4 12 12 ∗ Cm Cm Z P /nbc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 i ¯4 g 12 12 12 Z P (cid:48) /nb (cid:48) c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ g 12 12 12 Z P (cid:48) /nbc (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ∗ ¯4 ∗ g 12 12 12 ∗ Z P (cid:48) /n (cid:48) b (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 g 12 12 12 Z P /nb (cid:48) c (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C / Z , weak 4 ∗ i ∗ ¯4 g 12 12 12 ∗ Z 10 (000) 0 1 1 1 1 0 Z 01 (002) 1 1 1 0 0 1133.466 P (cid:48) /n (cid:48) bc (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 g 12 12 12 ∗ Z P c /nbc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i ∗ ¯4 ∗ g 12 12 ∗ Z P C /nbc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 12 ∗ i 12 12 ∗ ¯4 g ∗ Z P I /nbc Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i 12 12 12 ∗ 12 12 ∗ ¯4 g 12 12 12 Z P /nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 12 12 i ¯4 Cm Z P (cid:48) /nn (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 ∗ i ¯4 ∗ Cm Z P (cid:48) /nnm (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 i ∗ ¯4 ∗ g 12 12 ∗ Z P (cid:48) /n (cid:48) n (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 i ∗ ∗ ¯4 Cm Z P /nn (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C / Z , weak 4 12 12 ∗ i ∗ ¯4 g 12 12 ∗ Z 10 (000) 0 1 1 1 1 0 Z 01 (002) 1 0 1 1 0 0134.478 P (cid:48) /n (cid:48) nm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 ∗ i ∗ ¯4 g 12 12 ∗ Z P c /nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i ¯4 ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2134.481 P C /nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 ∗ 12 12 ∗ i 12 12 ∗ ¯4 g ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 0 2¯2134.482 P I /nnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ 12 12 i 12 12 12 ∗ 12 12 12 ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2135.483 P /mbc Classification = Z M X BS = Z SI = z +4 m, deco Z weak 4 12 12 i 12 12 12 ¯4 g 12 12 12 Cm Z P (cid:48) /mb (cid:48) c Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ 12 12 ∗ i 12 12 12 ¯4 ∗ g 12 12 12 Cm Z P (cid:48) /mbc (cid:48) Classification = Z M X BS = Z SI = z , . deco Z weak 4 ∗ 12 12 i 12 12 12 ∗ ¯4 ∗ g 12 12 12 ∗ Cm Z P (cid:48) /m (cid:48) b (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 i ∗ 12 12 12 ∗ ¯4 g 12 12 12 Z P /mb (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +2 m,π , z +4 m, deco Z , weak 4 12 12 ∗ i 12 12 12 ∗ ¯4 g 12 12 12 ∗ Cm Z 01 (000) 0 1 1 1 1 0 1¯100 Z 12 (00¯2) ¯1 1 1 0 0 ¯1 2011M(001; 0) 02 (000) 0 0 0 0 0 0 2¯200135.490 P (cid:48) /m (cid:48) bc (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ 12 12 ∗ i ∗ 12 12 12 ¯4 g 12 12 12 ∗ Z P c /mbc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | }∗ ∗ 12 12 i 12 12 ∗ ¯4 ∗ g 12 12 ∗ Cm Z P C /mbc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 }∗ 12 12 12 ∗ 12 12 12 i ∗ ¯4 12 12 12 ∗ m ∗ Cm Z ) 2 (000) 0 0 0 0 0 0 0 2¯200135.494 P I /mbc Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 12 }∗ 12 12 ∗ ∗ i 12 12 12 ¯4 12 12 ∗ g 12 12 12 Cm Z P /mnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 12 12 12 12 12 12 i ¯4 12 12 12 Cm Cm Z P (cid:48) /mn (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 12 12 12 ∗ 12 12 12 ∗ i ¯4 12 12 12 ∗ Cm Cm Z P (cid:48) /mnm (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 12 12 12 ∗ 12 12 12 i ∗ ¯4 12 12 12 ∗ m ∗ Cm Z P (cid:48) /m (cid:48) n (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 12 12 12 ∗ 12 12 12 i ∗ ∗ ¯4 12 12 12 Cm Z P /mn (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z +2 m,π , z +4 m, deco Z , weak 4 12 12 12 12 12 12 ∗ i ∗ ¯4 12 12 12 m ∗ Cm Z 01 (000) 0 1 1 1 1 0 1¯100 Z 12 (002) 1 0 1 1 0 0 2011M(001; 0) 02 (000) 0 0 0 0 0 0 2¯200136.502 P (cid:48) /m (cid:48) nm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 12 12 12 ∗ 12 12 12 ∗ i ∗ ¯4 12 12 12 m ∗ Z P c /mnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ 12 12 ∗ 12 12 ∗ i ¯4 12 12 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2 0000M(001; 0) 2 (000) 0 0 0 0 0 0 00 2¯200136.505 P C /mnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 12 12 12 12 12 ∗ i ∗ ¯4 12 12 12 g ∗ Cm Cm Z ) 2 (000) 0 0 0 0 0 0 0 2¯200 00M(1¯10; 0) 0 (000) 0 0 0 0 0 0 0 0000 2¯2136.506 P I /mnm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ ∗ i ¯4 ∗ Cm Cm Z P /nmc Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 i 12 12 12 ¯4 g Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200137.510 P (cid:48) /nm (cid:48) c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i 12 12 12 ¯4 ∗ g Z P (cid:48) /nmc (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i 12 12 12 ∗ ¯4 ∗ g ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200137.512 P (cid:48) /n (cid:48) m (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ 12 12 12 ∗ ¯4 g Z P /nm (cid:48) c (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C / Z , weak 4 ∗ i 12 12 12 ∗ ¯4 g ∗ Z 10 (000) 0 1 1 1 1 0 Z 11 (002) 1 0 0 1 1 1137.514 P (cid:48) /n (cid:48) mc (cid:48) Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ 12 12 12 ¯4 g ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200137.516 P c /nmc Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ i 12 12 ∗ ¯4 ∗ m ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 0 2¯200137.517 P C /nmc Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i 12 12 ∗ ∗ ¯4 12 12 ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯200137.518 P I /nmc Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ 12 12 ∗ i 12 12 12 ∗ 12 12 12 ¯4 g 12 12 ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 0 2¯200138.519 P /ncm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 i 12 12 ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2138.522 P (cid:48) /nc (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i 12 12 ¯4 ∗ Cm Z ) 0 (000) 0 0 0 0 0 2¯2138.523 P (cid:48) /ncm (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i 12 12 ∗ ¯4 ∗ m ∗ Z P (cid:48) /n (cid:48) c (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ 12 12 ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2138.525 P /nc (cid:48) m (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z C / Z , weak 4 ∗ i 12 12 ∗ ¯4 m ∗ Z 10 (000) 0 1 1 1 1 0 Z 01 (002) 1 0 1 1 0 0138.526 P (cid:48) /n (cid:48) cm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ 12 12 ¯4 m ∗ Z P c /ncm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ ∗ i 12 12 ¯4 ∗ Cm Z P C /ncm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ ∗ i 12 12 ∗ ∗ ¯4 12 12 ∗ g Cm Z ) 0 (000) 0 0 0 0 0 0 0 2¯2138.530 P I /ncm Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ ∗ i 12 12 12 ∗ 12 12 ¯4 Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2139.531 I /mmm Classification = Z M X BS = Z SI = z deco Z weak 4 i ¯4 Cm Cm Cm Z I (cid:48) /mm (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ Cm Cm Z I (cid:48) /mmm (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ∗ ¯4 ∗ m ∗ Cm Cm Z I (cid:48) /m (cid:48) m (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 Cm Z I /mm (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z +4 m, , z − m, deco Z , weak 4 ∗ i ∗ ¯4 m ∗ Cm Z 01 (000) 0 1 1 1 1 0 1¯1 Z 11 (11¯1) 0 1 1 1 1 0 20M(001; 0) 02 (000) 0 0 0 0 0 0 2¯2139.538 I (cid:48) /m (cid:48) mm (cid:48) Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 m ∗ Cm Z I c /mmm Classification = Z M X BS = Z SI = z deco Z weak { | 12 12 }∗ i ¯4 Cm Cm Cm Z I /mcm Classification = Z M X BS = Z SI = z +4 m, deco Z weak 4 i ¯4 Cm Cm Z ) 0 (000) 0 0 0 0 0 00 2¯2140.544 I (cid:48) /mc (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ ∗ i ¯4 ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 00 2¯2140.545 I (cid:48) /mcm (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ∗ i ∗ ¯4 ∗ g ∗ Cm Z I (cid:48) /m (cid:48) c (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ∗ i ∗ ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2140.547 I /mc (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z +4 m, , z − m, deco Z , weak 4 ∗ i ∗ ¯4 g ∗ Cm Z 01 (000) 0 1 1 1 1 0 1¯1 Z 11 (¯1¯11) 0 0 1 0 1 ¯1 20M(001; 0) 02 (000) 0 0 0 0 0 0 2¯2140.548 I (cid:48) /m (cid:48) cm (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ∗ ∗ i ∗ ¯4 g ∗ Z I c /mcm Classification = Z M X BS = Z SI = z +4 m, deco Z weak { | 12 12 }∗ ∗ i ∗ ¯4 m ∗ Cm Cm Z ) 0 (000) 0 0 0 0 0 0 0 00 2¯2141.551 I /amd Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ¯ 14 14 i ¯ 14 14 ¯4 ¯ 14 14 g 14 14 ¯ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2141.554 I (cid:48) /am (cid:48) d Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ¯ 14 14 ∗ ∗ i ¯ 14 14 ¯4 ¯ 14 14 ∗ g 14 14 ¯ Z I (cid:48) /amd (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 4 ¯ 14 14 ∗ i ¯ 14 14 ∗ ¯4 ¯ 14 14 ∗ g 14 14 ¯ ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2141.556 I (cid:48) /a (cid:48) m (cid:48) d Classification = Z X BS = Z SI = z ,S deco Z weak 4 ¯ 14 14 ∗ i ∗ ¯ 14 14 ∗ ¯4 ¯ 14 14 g 14 14 ¯ Z I /am (cid:48) d (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z (cid:48) P deco Z , weak 4 ¯ 14 14 ∗ i ¯ 14 14 ∗ ¯4 ¯ 14 14 g 14 14 ¯ ∗ Z 11 (000) 0 1 1 1 1 0 Z 01 (22¯2) 1 1 1 0 0 ¯1141.558 I (cid:48) /a (cid:48) md (cid:48) Classification = Z M X BS = Z SI = z ,S deco Z weak 4 ¯ 14 14 ∗ ∗ i ∗ ¯ 14 14 ¯4 ¯ 14 14 g 14 14 ¯ ∗ Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2141.560 I c /amd Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 14 14 14 ∗ 14 14 14 ∗ i ∗ ∗ ¯4 ¯ 14 14 Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2142.561 I /acd Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ¯ 14 14 i 14 14 14 ¯4 ¯ 14 14 g 14 14 14 Z I (cid:48) /ac (cid:48) d Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ¯ 14 14 ∗ ∗ i 14 14 14 ¯4 ¯ 14 14 ∗ g 14 14 14 Z I (cid:48) /acd (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 4 ¯ 14 14 ∗ i 14 14 14 ∗ ¯4 ¯ 14 14 ∗ g 14 14 14 ∗ Z I (cid:48) /a (cid:48) c (cid:48) d Classification = Z X BS = Z SI = z ,S deco Z weak 4 ¯ 14 14 ∗ i ∗ 14 14 14 ∗ ¯4 ¯ 14 14 g 14 14 14 Z I /ac (cid:48) d (cid:48) Classification = Z × Z X BS = Z , SI = z ,S , z (cid:48) P deco Z , weak 4 ¯ 14 14 ∗ i 14 14 14 ∗ ¯4 ¯ 14 14 g 14 14 14 ∗ Z 11 (000) 0 1 1 1 1 0 Z 10 (¯2¯22) ¯1 0 0 1 1 ¯1142.568 I (cid:48) /a (cid:48) cd (cid:48) Classification = Z X BS = Z SI = z ,S deco Z weak 4 ¯ 14 14 ∗ ∗ i ∗ 14 14 14 ¯4 ¯ 14 14 g 14 14 14 ∗ Z I c /acd Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 }∗ 14 14 14 ∗ 14 14 14 i ∗ ¯4 ¯ 14 14 g Z P Z X BS = Z SI = z C deco Z weak 3 Z P ¯3 Classification = Z × Z X BS = Z , SI = z C , z P deco Z , weak 3 iZ 02 (000) 0 1 Z 50 (00¯1) 0 0147.16 P c ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ iZ R ¯3 Classification = Z × Z X BS = Z , SI = z P, , z P deco Z , weak 3 iZ 02 (000) 0 1 Z 10 (111) 0 0148.20 R I ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ iZ P (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 3 ∗ Z Z P (cid:48) Z × Z X BS = Z SI = z C deco Z weak 3 ∗ Z Z P m (cid:48) Z X BS = Z SI = z C deco Z weak 3 m ¯210 ∗ Z P m (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 3 m ∗ Z P c (cid:48) Z X BS = Z SI = z C deco Z weak 3 g ¯21000 ∗ Z P c (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 3 g ∗ Z P ¯31 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i Cm Z P ¯31 m (cid:48) Classification = Z × Z X BS = Z , SI = z C , z P deco Z , weak 3 ∗ iZ 02 (000) 0 1 1 Z 10 (001) 0 0 0162.78 P c ¯31 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Z P ¯31 c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 iZ P ¯31 c (cid:48) Classification = Z × Z X BS = Z , SI = z C,π / , z P deco Z , weak 3 ∗ iZ 02 (000) 0 1 1 Z 22 (00¯2) 0 0 1163.84 P c ¯31 c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ iZ P ¯3 m Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i Cm Z ) 0 (000) 0 0 0 2¯2164.89 P ¯3 m (cid:48) Z × Z X BS = Z , SI = z C , z P deco Z , weak 3 ∗ iZ 02 (000) 0 1 1 Z 10 (001) 0 0 0164.90 P c ¯3 m Z M X BS = Z SI = z (cid:48) P deco Z weak { | }∗ i Cm Z ) 0 (000) 0 0 0 0 2¯2165.91 P ¯3 c Z X BS = Z SI = z (cid:48) P deco Z weak 3 iZ P ¯3 c (cid:48) Z × Z X BS = Z , SI = z C,π / , z P deco Z , weak 3 ∗ iZ 02 (000) 0 1 1 Z 12 (002) 0 0 1165.96 P c ¯3 c Z X BS = Z SI = z (cid:48) P deco Z weak { | }∗ ∗ iZ R ¯3 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i Cm ¯210(2) Z R ¯3 m (cid:48) Classification = Z × Z X BS = Z , SI = z P, , z P deco Z , weak 3 ∗ iZ 02 (000) 0 1 1 Z 10 (¯1¯1¯1) 0 0 0166.102 R I ¯3 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm ¯210(2) Z R ¯3 c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 iZ R ¯3 c (cid:48) Classification = Z × Z X BS = Z SI = z P deco Z weak 3 ∗ iZ Z R I ¯3 c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ iZ P Z X BS = Z SI = z C deco Z weak 6 Z P Classification = Z X BS = Z SI = z C deco Z weak 6 Z P Classification = Z X BS = Z SI = z C deco Z weak 6 Z P Classification = Z X BS = Z SI = z C deco Z weak 6 Z P ¯6 Classification = Z M × Z X BS = Z , , SI = z +3 m, , z − m, , z +3 m,π deco Z , , weak ¯6 Cm Z 120 (000) 1 1¯100 Z 100 (001) 0 1001M(001; 0) 121 (000) 0 1¯11¯1M(001; ) 122 (000) 0 1¯1¯11 P c ¯6 Classification = Z M X BS = Z SI = z +3 m, deco Z weak { | }∗ ¯6 Cm Z P /m Classification = Z M × Z X BS = Z , , SI = z +6 m, , z − m, , z +6 m,π deco Z , , weak 6 i Cm Z 150 (000) 0 1 1¯100 Z 100 (001) 0 0 1001M(001; 0) 151 (000) 0 0 1¯11¯1M(001; ) 155 (000) 0 0 1¯1¯11175.139 P (cid:48) /m Classification = Z M X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak 6 ∗ i ∗ Cm Z 10 (000) 1 0 1¯100M(001; 0) 11 (000) 0 0 1¯11¯1M(001; ) 12 (000) 0 0 1¯1¯11175.141 P (cid:48) /m (cid:48) Classification = Z X BS = Z , SI = z C / , z P deco Z , weak 6 ∗ iZ 02 (000) 1 1175.142 P c /m Classification = Z M X BS = Z SI = z +6 m, deco Z weak { | }∗ i Cm Z P /m Classification = Z M × Z X BS = Z , SI = z +3 m,π , z +6 m, deco Z , weak 6 i Cm Z 01 (000) 0 1 1¯100 Z 12 (002) 1 0 2011M(001; ) 02 (000) 0 0 2¯200176.145 P (cid:48) /m Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ i ∗ Cm Z ) 2 (000) 0 0 2¯200176.147 P (cid:48) /m (cid:48) Classification = Z X BS = Z SI = z P deco Z weak 6 ∗ iZ P c /m Classification = Z M X BS = Z SI = z +6 m, deco Z weak { | }∗ ∗ i ∗ Cm Z ) 2 (000) 0 0 0 2¯200177.153 P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 6 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 6 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 6 ∗ ∗ Z Z P (cid:48) (cid:48) Classification = Z × Z X BS = Z SI = z C deco Z weak 6 ∗ ∗ Z Z P m (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 6 m ¯120 ∗ m ∗ Z P c (cid:48) c (cid:48) Classification = Z X BS = Z SI = z (cid:48) C deco Z weak 6 g ¯12000 ∗ g ∗ Z P c (cid:48) m (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 6 g ¯12000 ∗ m ∗ Z P m (cid:48) c (cid:48) Classification = Z X BS = Z SI = z C deco Z weak 6 m ¯120 ∗ g ∗ Z P ¯6 m Z M X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak ¯6 Cm Cm Z 10 (000) 1 0 1¯100 1¯1M(001; 0) 11 (000) 0 0 1¯11¯1 00M(001; ) 12 (000) 0 0 1¯1¯11 00M(110; ) 00 (000) 0 0 0000 2¯2187.213 P ¯6 m (cid:48) (cid:48) Classification = Z M × Z X BS = Z , , SI = z +3 m, , z − m, , z +3 m,π deco Z , , weak ¯6 m ¯120 ∗ ∗ Cm Z 120 (000) 1 0 1 1¯100 Z 100 (001) 0 0 0 1001M(001; 0) 121 (000) 0 0 0 1¯11¯1M(001; ) 122 (000) 0 0 0 1¯1¯11187.214 P c ¯6 m Z M X BS = Z SI = z +3 m, deco Z weak { | }∗ ¯6 Cm Cm Z ) 0 (000) 0 0 0 0000 2¯2188.215 P ¯6 c Z M X BS = Z SI = z +3 m, deco Z weak ¯6 g ¯12000 Cm Z ) 2 (000) 0 0 0 2¯200188.219 P ¯6 c (cid:48) (cid:48) Classification = Z M × Z X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak ¯6 g ¯12000 ∗ ∗ Cm Z 10 (000) 1 0 1 1¯100 Z 21 (00¯2) 1 ¯1 0 2011M(001; ) 20 (000) 0 0 0 2¯200188.220 P c ¯6 c Z M X BS = Z SI = z +3 m, deco Z weak { | }∗ ¯6 ∗ m ¯120 ∗ Cm Z ) 2 (000) 0 0 0 0 2¯200189.221 P ¯62 m Classification = Z M X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak ¯6 Cm Cm Z 10 (000) 1 0 1¯1 1¯100M(100; 0) 00 (000) 0 0 2¯2 0000M(001; 0) 11 (000) 0 0 00 1¯11¯1M(001; ) 12 (000) 0 0 00 1¯1¯11189.225 P ¯62 (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , , SI = z +3 m, , z − m, , z +3 m,π deco Z , , weak ¯6 ∗ m ∗ Cm Z 120 (000) 1 1 0 1¯100 Z 100 (001) 0 0 0 1001M(001; 0) 121 (000) 0 0 0 1¯11¯1M(001; ) 122 (000) 0 0 0 1¯1¯11189.226 P c ¯62 m Classification = Z M X BS = Z SI = z +3 m, deco Z weak { | }∗ ¯6 Cm Cm Z P ¯62 c Classification = Z M X BS = Z SI = z +3 m, deco Z weak ¯6 g Cm Z ) 2 (000) 0 0 0 2¯200190.231 P ¯62 (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak ¯6 ∗ g ∗ Cm Z 10 (000) 1 1 0 1¯100 Z 21 (002) 1 0 1 2011M(001; ) 20 (000) 0 0 0 2¯200190.232 P c ¯62 c Classification = Z M X BS = Z SI = z +3 m, deco Z weak { | }∗ ¯6 ∗ m ∗ Cm Z ) 2 (000) 0 0 0 0 2¯200191.233 P /mmm Classification = Z M X BS = Z , SI = z +6 m,π , z deco Z , weak 6 i Cm Cm Cm ¯210(2) Z 07 (000) 0 0 1 0 1¯1 1¯100 1¯1M(100; 0) 06 (000) 0 0 0 0 2¯2 0000 00M(001; 0) 12 (000) 0 0 0 0 00 1¯11¯1 00M(001; ) 50 (000) 0 0 0 0 00 1¯1¯11 00M(¯210; 0) 06 (000) 0 0 0 0 00 0000 2¯2191.236 P (cid:48) /mm (cid:48) m Classification = Z M X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak 6 ∗ i ∗ ∗ Cm Cm Z 10 (000) 1 0 0 1 1¯1 1¯100M(100; 0) 00 (000) 0 0 0 0 2¯2 0000M(001; 0) 11 (000) 0 0 0 0 00 1¯11¯1M(001; ) 12 (000) 0 0 0 0 00 1¯1¯11191.237 P (cid:48) /mmm (cid:48) Classification = Z M X BS = Z , SI = z +3 m, , z +3 m,π deco Z , weak 6 ∗ ∗ i ∗ m ∗ Cm Cm ¯210(0) Z 10 (000) 1 1 0 0 0 1¯100 1¯1M(001; 0) 11 (000) 0 0 0 0 0 1¯11¯1 00M(001; ) 12 (000) 0 0 0 0 0 1¯1¯11 00M(¯210; 0) 00 (000) 0 0 0 0 0 0000 2¯2191.238 P (cid:48) /m (cid:48) m (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ ∗ i Cm Z P (cid:48) /m (cid:48) mm (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ i ∗ m ∗ Cm ¯210(2) Z P /mm (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , , SI = z +6 m, , z − m, , z +6 m,π deco Z , , weak 6 ∗ i ∗ m ∗ Cm Z 150 (000) 0 1 1 1 0 1¯100 Z 100 (00¯1) 0 0 0 0 0 1001M(001; 0) 151 (000) 0 0 0 0 0 1¯11¯1M(001; ) 155 (000) 0 0 0 0 0 1¯1¯11191.242 P c /mmm Classification = Z M X BS = Z SI = z deco Z weak { | }∗ i Cm Cm Cm ¯210(2) Z P /mcc Classification = Z M X BS = Z SI = z +6 m, deco Z weak 6 i g Cm Z ) 2 (000) 0 0 0 0 0 2¯200192.246 P (cid:48) /mc (cid:48) c Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ i ∗ ∗ g Cm Z ) 2 (000) 0 0 0 0 0 2¯200192.247 P (cid:48) /mcc (cid:48) Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ ∗ i ∗ g ∗ Cm Z ) 2 (000) 0 0 0 0 0 2¯200192.248 P (cid:48) /m (cid:48) c (cid:48) c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ ∗ i g Z P (cid:48) /m (cid:48) cc (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ i ∗ g ∗ Z P /mc (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +6 m, , z +6 m,π deco Z , weak 6 ∗ i ∗ g ∗ Cm Z 10 (000) 0 1 1 1 0 1¯100 Z 21 (00¯2) 0 0 1 0 ¯1 2011M(001; ) 20 (000) 0 0 0 0 0 2¯200192.252 P c /mcc Classification = Z M X BS = Z SI = z +6 m, deco Z weak { | }∗ ∗ i ∗ m ∗ Cm Z P /mcm Classification = Z M X BS = Z SI = z +6 m, deco Z weak 6 i Cm Cm Z ) 2 (000) 0 0 0 0 00 2¯200193.256 P (cid:48) /mc (cid:48) m Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ i ∗ ∗ Cm Cm Z ) 2 (000) 0 0 0 0 00 2¯200193.257 P (cid:48) /mcm (cid:48) Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ ∗ i ∗ m ∗ Cm Z ) 2 (000) 0 0 0 0 0 2¯200193.258 P (cid:48) /m (cid:48) c (cid:48) m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ ∗ i Cm Z P (cid:48) /m (cid:48) cm (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ i ∗ m ∗ Z P /mc (cid:48) m (cid:48) Classification = Z M × Z X BS = Z , SI = z +3 m,π , z +6 m, deco Z , weak 6 ∗ i ∗ m ∗ Cm Z 21 (000) 0 1 1 1 0 1¯100 Z 02 (00¯2) ¯1 1 0 0 0 2011 M(001; ) 12 (000) 0 0 0 0 0 2¯200193.262 P c /mcm Classification = Z M X BS = Z SI = z +6 m, deco Z weak { | }∗ ∗ i ∗ ∗ Cm Cm Z ) 2 (000) 0 0 0 0 0 00 2¯200194.263 P /mmc Classification = Z M X BS = Z SI = z +6 m, deco Z weak 6 i g Cm Cm Z ) 2 (000) 0 0 0 0 0 00 2¯200194.266 P (cid:48) /mm (cid:48) c Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ i ∗ ∗ g Cm Z ) 2 (000) 0 0 0 0 0 2¯200194.267 P (cid:48) /mmc (cid:48) Classification = Z M X BS = Z SI = z +3 m, deco Z weak 6 ∗ ∗ i ∗ g ∗ Cm Cm Z ) 2 (000) 0 0 0 0 0 00 2¯200194.268 P (cid:48) /m (cid:48) m (cid:48) c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ ∗ i g Z P (cid:48) /m (cid:48) mc (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 6 ∗ i ∗ g ∗ Cm Z P /mm (cid:48) c (cid:48) Classification = Z M × Z X BS = Z , SI = z +3 m,π , z +6 m, deco Z , weak 6 ∗ i ∗ g ∗ Cm Z 21 (000) 0 1 1 1 0 1¯100 Z 02 (00¯2) ¯1 0 0 1 ¯1 2011M(001; ) 12 (000) 0 0 0 0 0 2¯200194.272 P c /mmc Classification = Z M X BS = Z SI = z +6 m, deco Z weak { | }∗ ∗ ∗ i ∗ m ∗ Cm Cm ¯210(2) Z ) 2 (000) 0 0 0 0 0 0 2¯200 00M(¯210; 0) 0 (000) 0 0 0 0 0 0 0000 2¯2200.14 P m ¯3 Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 3 i Cm Z 03 (000) 0 0 1 1¯100M(100; 0) 12 (000) 0 0 0 1¯11¯1M(100; ) 10 (000) 0 0 0 1¯1¯11200.17 P I m ¯3 Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm Z P n ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 12 12 iZ P I n ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 12 12 i 12 12 12 ∗ Z F m ¯3 Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i Cm Z F S m ¯3 Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i Cm Z F d ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 14 14 iZ F S d ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 14 14 iZ Im ¯3 Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i Cm Z P a ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 12 12 iZ P I a ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i 12 12 12 ∗ Z Ia ¯3 Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 iZ P ¯43 m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ¯4 Cm Z P I ¯43 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 Cm Z F ¯43 m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ¯4 Cm Z ) 0 (000) 0 0 2¯2216.77 F S ¯43 m Classification = Z M X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 Cm Z I ¯43 m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ¯4 Cm Z P ¯43 n Classification = Z X BS = Z SI = z ,S deco Z weak 3 ¯4 12 12 12 g 12 12 12 Z P I ¯43 n Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 ∗ m ∗ Z F ¯43 c Classification = Z X BS = Z SI = z ,S deco Z weak 3 ¯4 12 12 12 g 12 12 12 Z F S ¯43 c Classification = Z X BS = Z SI = z ,S deco Z weak { | 12 12 12 }∗ ¯4 ∗ m ∗ Z I ¯43 d Classification = Z X BS = Z SI = z ,S deco Z weak 3 ¯4 14 14 ¯ g 14 14 14 Z P m ¯3 m Classification = Z M X BS = Z , SI = z +4 m,π , z deco Z , weak 3 i ¯4 Cm Cm Z 03 (000) 0 0 1 0 1 1¯100 1¯1M(100; 0) 12 (000) 0 0 0 0 0 1¯11¯1 00M(100; ) 30 (000) 0 0 0 0 0 1¯1¯11 00M(0¯11; 0) 04 (000) 0 0 0 0 0 0000 2¯2221.94 P m (cid:48) ¯3 (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ∗ i ∗ ∗ ¯4 Cm Z P m ¯3 m (cid:48) Classification = Z M X BS = Z , SI = z (cid:48) P, , z (cid:48) P deco Z , weak 3 ∗ i ∗ ¯4 ∗ m ∗ Cm Z 03 (000) 0 1 1 1 0 0 1¯100M(100; 0) 12 (000) 0 0 0 0 0 0 1¯11¯1M(100; ) 10 (000) 0 0 0 0 0 0 1¯1¯11221.97 P I m ¯3 m Classification = Z M X BS = Z SI = z deco Z weak { | 12 12 12 }∗ i ¯4 Cm Cm Z P n ¯3 n Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 i ¯4 g Z P n (cid:48) ¯3 (cid:48) n Classification = Z X BS = Z SI = z ,S deco Z weak 3 ∗ i ∗ ∗ ¯4 g Z P n ¯3 n (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 ∗ i ∗ ¯4 ∗ g ∗ Z P I n ¯3 n Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i 12 12 12 ∗ ¯4 ∗ g ∗ Z P m ¯3 n Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 12 12 12 i 12 12 12 ¯4 12 12 12 g 12 12 12 Cm Z P m (cid:48) ¯3 (cid:48) n Classification = Z X BS = Z SI = z ,S deco Z weak 3 12 12 12 ∗ i ∗ 12 12 12 ∗ ¯4 12 12 12 g 12 12 12 Z P m ¯3 n (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 12 12 12 ∗ i 12 12 12 ∗ ¯4 12 12 12 ∗ g 12 12 12 ∗ Cm Z P I m ¯3 n Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i ∗ ¯4 ∗ m ∗ Cm Z P n ¯3 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i 12 12 ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2224.112 P n (cid:48) ¯3 (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ∗ i ∗ 12 12 ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2224.113 P n ¯3 m (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 ∗ i 12 12 ∗ ¯4 ∗ g ∗ Z P I n ¯3 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i 12 12 12 ∗ ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2225.116 F m ¯3 m Classification = Z M X BS = Z SI = z deco Z weak 3 i ¯4 Cm Cm Z ) 2 (000) 0 0 0 0 0 00 2¯2225.118 F m (cid:48) ¯3 (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ∗ i ∗ ∗ ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2225.119 F m ¯3 m (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 ∗ i ∗ ¯4 ∗ m ∗ Cm Z F S m ¯3 m Classification = Z M X BS = Z SI = z deco Z weak { | 12 12 12 }∗ i ¯4 Cm Cm Z F m ¯3 c Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 12 12 12 i 12 12 12 ¯4 12 12 12 g 12 12 12 Cm Z F m (cid:48) ¯3 (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 3 12 12 12 ∗ i ∗ 12 12 12 ∗ ¯4 12 12 12 g 12 12 12 Z F m ¯3 c (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 12 12 12 ∗ i 12 12 12 ∗ ¯4 12 12 12 ∗ g 12 12 12 ∗ Cm Z F S m ¯3 c Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ ∗ i ∗ ¯4 ∗ m ∗ Cm Z F d ¯3 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 i 14 14 ¯4 Cm Z ) 0 (000) 0 0 0 0 0 2¯2227.130 F d (cid:48) ¯3 (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ∗ i ∗ 14 14 ∗ ¯4 Cm Z M(011; ) 0 (000) 0 0 0 0 0 2¯2227.131 F d ¯3 m (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 ∗ i 14 14 ∗ ¯4 ∗ g ∗ Z F S d ¯3 m Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ i 14 14 ¯4 Cm Z ) 0 (000) 0 0 0 0 0 0 2¯2228.134 F d ¯3 c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 14 12 14 i 14 14 12 ¯4 14 12 14 g 14 12 14 Z F d (cid:48) ¯3 (cid:48) c Classification = Z X BS = Z SI = z ,S deco Z weak 3 14 12 14 ∗ i ∗ 14 14 12 ∗ ¯4 14 12 14 g 14 12 14 Z F d ¯3 c (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 14 12 14 ∗ i 14 14 12 ∗ ¯4 14 12 14 ∗ g 14 12 14 ∗ Z F S d ¯3 c Classification = Z X BS = Z SI = z (cid:48) P deco Z weak { | 12 12 12 }∗ 14 14 14 14 ∗ i 14 14 ∗ ¯4 14 14 ∗ m ∗ Z Im ¯3 m Classification = Z M X BS = Z SI = z deco Z weak 3 i ¯4 Cm Cm Z Im (cid:48) ¯3 (cid:48) m Classification = Z M X BS = Z SI = z ,S deco Z weak 3 ∗ i ∗ ∗ ¯4 Cm Z Im ¯3 m (cid:48) Classification = Z M X BS = Z SI = z (cid:48) P deco Z weak 3 ∗ i ∗ ¯4 ∗ m ∗ Cm Z Ia ¯3 d Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 14 14 ¯ i 14 14 14 ¯4 14 14 ¯ g 14 14 ¯ Z Ia (cid:48) ¯3 (cid:48) d Classification = Z X BS = Z SI = z ,S deco Z weak 3 14 14 ¯ ∗ i ∗ 14 14 14 ∗ ¯4 14 14 ¯ g 14 14 ¯ Z Ia ¯3 d (cid:48) Classification = Z X BS = Z SI = z (cid:48) P deco Z weak 3 14 14 ¯ ∗ i 14 14 14 ∗ ¯4 14 14 ¯ ∗ g 14 14 ¯ ∗ Z Appendix P: Table: Invariants of TCI classificationgenerators for MSGs with trivial SI group TABLE XV: MSGs with trivial SI group1.1 P Z deco weak Z (00¯1) Z (010) Z (¯100) 1.3 P S Z deco weak { | }∗ Z (000) 12.6 P ¯1 (cid:48) Classification = N/Adeco weak i ∗ P (cid:48) Classification = Z × Z deco weak 2 ∗ Z (000) 1 Z (00¯1) 0 Z (¯100) 03.5 P b Z deco weak { | }∗ Z (000) 1 03.6 P C Z deco weak { | 12 12 }∗ Z (000) 1 04.7 P Classification = Z deco weak 2 Z (020) 14.9 P (cid:48) Classification = Z × Z deco weak 2 ∗ Z (000) 1 Z (¯100) 0 Z (00¯1) 04.10 P a Classification = Z deco weak { | }∗ Z (000) 1 04.11 P b Classification = Z deco weak { | }∗ ∗ Z (000) 1 14.12 P C Classification = Z deco weak { | 12 12 }∗ ∗ Z (000) 1 15.13 C Z deco weak 2 Z (110) 05.15 C (cid:48) Classification = Z × Z deco weak 2 ∗ Z (000) 1 Z (¯110) 0 Z (001) 05.16 C c Z deco weak { | }∗ Z (000) 1 05.17 C a Z deco weak { | 12 12 }∗ Z (000) 1 06.18 P m Classification = Z M × Z deco weak Cm Z (000) 1¯100 Z (0¯10) 1001M(010; 0) (000) 1¯11¯1M(010; ) (000) 1¯1¯116.20 P m (cid:48) Classification = Z deco weak m ∗ Z (00¯1) 0 Z (¯100) 0 P a m Classification = Z M deco weak { | }∗ Cm Z (000) 1 1¯100M(010; 0) (000) 0 1¯11¯1M(010; ) (000) 0 1¯1¯116.22 P b m Classification = Z M deco weak { | }∗ Cm Z (000) 1 1¯100M(010; 0) (000) 0 2¯2006.23 P C m Classification = Z M deco weak { | 12 12 }∗ Cm Z (000) 1 1¯100M(010; 0) (000) 0 2¯2007.24 P c Classification = Z × Z deco weak g Z (000) 1 Z (010) 07.26 P c (cid:48) Classification = Z deco weak g ∗ Z (¯100) 0 Z (002) 17.27 P a c Classification = Z deco weak { | }∗ g Z (000) 1 17.28 P c c Classification = Z deco weak { | }∗ m ∗ Z (000) 1 07.29 P b c Classification = Z deco weak { | }∗ g Z (000) 1 17.30 P C c Classification = Z deco weak { | 12 12 }∗ g Z (000) 1 17.31 P A c Classification = Z deco weak { | 12 12 }∗ g ∗ Z (000) 1 08.32 Cm Classification = Z M × Z deco weak Cm Z (000) 1¯1 Z (110) 20M(010; 0) (000) 2¯28.34 Cm (cid:48) Classification = Z deco weak m ∗ Z (00¯1) 0 Z (1¯10) 08.35 C c m Classification = Z M deco weak { | }∗ Cm Z (000) 1 1¯1M(010; 0) (000) 0 2¯28.36 C a m Classification = Z M deco weak { | 12 12 }∗ Cm Z (000) 1 1¯1M(010; 0) (000) 0 2¯29.37 Cc Classification = Z × Z deco weak g Z (000) 1 Z (110) 09.39 Cc (cid:48) Classification = Z deco weak g ∗ Z (¯110) 0 Z (002) 19.40 C c c Classification = Z deco weak { | }∗ m ∗ Z (000) 1 09.41 C a c Classification = Z deco weak { | 12 12 }∗ g Z (000) 1 110.44 P (cid:48) /m Classification = Z M deco weak 2 ∗ i ∗ Cm Z (000) 1 0 1¯100M(010; 0) (000) 0 0 1¯11¯1M(010; ) (000) 0 0 1¯1¯1110.45 P /m (cid:48) Classification = N/Adeco weak 2 i ∗ P (cid:48) /m Classification = Z M deco weak 2 ∗ i ∗ Cm Z (000) 1 0 1¯100M(010; ) (000) 0 0 2¯20011.53 P /m (cid:48) Classification = N/Adeco weak 2 i ∗ C (cid:48) /m Classification = Z M deco weak 2 ∗ i ∗ Cm Z (000) 1 0 1¯1M(010; 0) (000) 0 0 2¯212.61 C /m (cid:48) Classification = N/Adeco weak 2 i ∗ P (cid:48) /c Classification = Z deco weak 2 ∗ i ∗ Z (000) 1 013.68 P /c (cid:48) Classification = N/Adeco weak 2 i ∗ P (cid:48) /c Classification = Z deco weak 2 12 12 ∗ i ∗ Z (000) 1 014.78 P /c (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ C (cid:48) /c Classification = Z deco weak 2 ∗ i ∗ Z (000) 1 015.88 C /c (cid:48) Classification = N/Adeco weak 2 i ∗ P 222 Classification = N/Adeco weak 2 P a 222 Classification = Z deco weak { | }∗ Z (000) 1 0 0 016.5 P C 222 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 016.6 P I 222 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0 017.7 P Classification = N/Adeco weak 2 P (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ ∗ Z (000) 0 1 1 Z (002) 1 0 117.11 P a Classification = Z deco weak { | }∗ Z (000) 1 0 0 017.12 P c Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 1 017.13 P B Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 1 017.14 P C Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 017.15 P I Classification = Z deco weak { | 12 12 12 }∗ 12 12 ∗ Z (000) 1 1 0 018.16 P 12 12 12 12 P (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ 12 12 ∗ 12 12 Z (000) 1 1 0 Z (200) 0 1 118.20 P b Z deco weak { | }∗ 12 12 ∗ ∗ Z (000) 1 0 1 118.21 P c Z deco weak { | }∗ 12 12 12 12 Z (000) 1 0 0 018.22 P B Z deco weak { | }∗ 12 12 ∗ 12 12 ∗ Z (000) 1 1 1 018.23 P C Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 0 1 118.24 P I Z deco weak { | 12 12 12 }∗ ∗ ∗ Z (000) 1 0 1 119.25 P Classification = N/Adeco weak 2 12 12 12 12 P (cid:48) (cid:48) Classification = Z × Z deco weak 2 12 12 ∗ 12 12 ∗ Z (000) 0 1 1 Z (002) 1 0 119.28 P c Classification = Z deco weak { | }∗ ∗ ∗ 12 12 Z (000) 1 1 1 019.29 P C Classification = Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 0 1 119.30 P I Classification = Z deco weak { | 12 12 12 }∗ ∗ ∗ ∗ Z (000) 1 1 1 120.31 C Classification = N/Adeco weak 2 C (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ ∗ Z (000) 0 1 1 Z (002) 1 1 020.34 C (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ ∗ Z (000) 1 1 0 Z (1¯10) 1 1 020.35 C c Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 1 020.36 C a Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 020.37 C A Classification = Z deco weak { | 12 12 12 }∗ ∗ ∗ Z (000) 1 1 1 021.38 C 222 Classification = N/Adeco weak 2 C (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ ∗ Z (000) 1 1 0 Z (1¯10) 0 0 021.42 C c 222 Classification = Z deco weak { | }∗ Z (000) 1 0 0 021.43 C a 222 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 021.44 C A 222 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0 022.45 F 222 Classification = N/Adeco weak 2 F (cid:48) (cid:48) Z × Z deco weak 2 ∗ ∗ Z (000) 0 1 1 Z (110) 0 0 022.48 F S 222 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0 023.49 I 222 Classification = N/Adeco weak 2 I (cid:48) (cid:48) Z × Z deco weak 2 ∗ ∗ Z (000) 0 1 1 Z (11¯1) 0 0 023.52 I c 222 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 024.53 I Classification = N/Adeco weak 2 I (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ ∗ Z (000) 0 1 1 Z (¯1¯11) 0 1 024.56 I c Classification = Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 1 0 125.57 P mm Z M deco weak 2 Cm Cm Z (000) 0 1¯100 1¯100M(100; 0) (000) 0 1¯11¯1 0000M(100; ) (000) 0 1¯1¯11 0000M(010; 0) (000) 0 0000 1¯11¯1M(010; ) (000) 0 0000 1¯1¯1125.59 P m (cid:48) m (cid:48) Classification = Z M × Z deco weak 2 ∗ m ∗ Cm Z (000) 1 0 1¯100 Z (0¯10) 0 0 1001M(010; 0) (000) 0 0 1¯11¯1M(010; ) (000) 0 0 1¯1¯1125.61 P c mm Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯100 1¯100M(100; 0) (000) 0 0 1¯11¯1 0000M(100; ) (000) 0 0 1¯1¯11 0000M(010; 0) (000) 0 0 0000 1¯11¯1M(010; ) (000) 0 0 0000 1¯1¯1125.62 P a mm Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯100 1¯100M(100; 0) (000) 0 0 2¯200 0000M(010; 0) (000) 0 0 0000 1¯11¯1M(010; ) (000) 0 0 0000 1¯1¯1125.63 P C mm Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯100 1¯100M(010; 0) (000) 0 0 2¯200 0000M(100; 0) (000) 0 0 0000 2¯20025.64 P A mm Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯100 1¯100M(100; 0) (000) 0 0 1¯11¯1 0000M(100; ) (000) 0 0 1¯1¯11 0000M(010; ) (000) 0 0 0000 2¯20025.65 P I mm Z M deco weak { | 12 12 12 }∗ Cm Cm Z (000) 1 0 1¯100 1¯100M(010; 0) (000) 0 0 2¯200 0000 M(100; 0) (000) 0 0 0000 2¯20026.66 P mc Classification = Z M deco weak 2 g Cm Z (000) 0 1 1¯100M(100; 0) (000) 0 0 1¯11¯1M(100; ) (000) 0 0 1¯1¯1126.68 P m (cid:48) c (cid:48) Classification = Z × Z deco weak 2 ∗ g m ∗ Z (000) 1 1 0 Z (010) 0 0 026.69 P mc (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯100 Z (¯100) 0 0 1001M(100; 0) (000) 0 0 1¯11¯1M(100; ) (000) 0 0 1¯1¯1126.70 P m (cid:48) c (cid:48) Classification = Z deco weak 2 g ∗ m ∗ Z (002) 1 1 026.71 P a mc Classification = Z M deco weak { | }∗ g Cm Z (000) 1 0 1 1¯100M(100; 0) (000) 0 0 0 2¯20026.72 P b mc Classification = Z M deco weak { | }∗ g Cm Z (000) 1 0 1 1¯100M(100; 0) (000) 0 0 0 1¯11¯1M(100; ) (000) 0 0 0 1¯1¯1126.73 P c mc Classification = Z M deco weak { | }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯100M(100; 0) (000) 0 0 0 1¯11¯1M(100; ) (000) 0 0 0 1¯1¯1126.74 P A mc Classification = Z M deco weak { | 12 12 }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯100M(100; 0) (000) 0 0 0 1¯11¯1M(100; ) (000) 0 0 0 1¯1¯1126.75 P B mc Classification = Z M deco weak { | }∗ ∗ g Cm Z (000) 1 1 1 1¯100M(100; ) (000) 0 0 0 2¯20026.76 P C mc Classification = Z M deco weak { | 12 12 }∗ g Cm Z (000) 1 0 1 1¯100M(100; ) (000) 0 0 0 2¯20026.77 P I mc Classification = Z M deco weak { | 12 12 12 }∗ 12 12 ∗ g 12 12 ∗ Cm Z (000) 1 1 0 1¯100M(100; ) (000) 0 0 0 2¯20027.78 P cc Z deco weak 2 g g Z (000) 0 1 127.80 P c (cid:48) c (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (0¯10) 0 0 027.82 P c cc Z deco weak { | }∗ m ∗ m ∗ Z (000) 1 0 0 027.83 P a cc Z deco weak { | }∗ g g Z (000) 1 0 1 127.84 P C cc Z deco weak { | 12 12 }∗ g g Z (000) 1 0 1 127.85 P A cc Z deco weak { | 12 12 }∗ g g Z (000) 1 0 1 127.86 P I cc Z deco weak { | 12 12 12 }∗ g g Z (000) 1 0 1 128.87 P ma Z M deco weak 2 g Cm Z (000) 0 1 1¯100M(100; ) (000) 0 0 2¯20028.89 P m (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (0¯10) 0 0 028.90 P ma (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯100 Z (¯200) 0 ¯1 2011M(100; ) (000) 0 0 2¯20028.92 P a ma Z M deco weak { | }∗ m ∗ Cm Z (000) 1 0 0 1¯100M(100; ) (000) 0 0 0 2¯20028.93 P b ma Z M deco weak { | }∗ g Cm Z (000) 1 0 1 1¯100M(100; ) (000) 0 0 0 2¯20028.94 P c ma Z M deco weak { | }∗ g Cm Z (000) 1 0 1 1¯100M(100; ) (000) 0 0 0 2¯20028.95 P A ma Z M deco weak { | 12 12 }∗ g Cm Z (000) 1 0 1 1¯100M(100; ) (000) 0 0 0 2¯20028.96 P B ma Z M deco weak { | }∗ g ∗ Cm Z (000) 1 0 0 1¯100M(100; ) (000) 0 0 0 2¯20028.97 P C ma Z M deco weak { | 12 12 }∗ 12 12 ∗ g ∗ Cm Z (000) 1 1 0 1¯100 M(100; ) (000) 0 0 0 2¯20028.98 P I ma Z M deco weak { | 12 12 12 }∗ g Cm Z (000) 1 0 1 1¯100M(100; ) (000) 0 0 0 2¯20029.99 P ca Classification = Z deco weak 2 g g Z (000) 0 1 129.101 P c (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (0¯10) 0 0 029.102 P ca (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ g ∗ g Z (000) 1 0 1 Z (¯200) 1 ¯1 029.103 P c (cid:48) a (cid:48) Classification = Z deco weak 2 g ∗ g ∗ Z (002) 1 0 129.104 P a ca Classification = Z deco weak { | }∗ g ∗ m ∗ Z (000) 1 0 0 029.105 P b ca Classification = Z deco weak { | }∗ g g Z (000) 1 0 1 129.106 P c ca Classification = Z deco weak { | }∗ ∗ g g ∗ Z (000) 1 1 1 029.107 P A ca Classification = Z deco weak { | 12 12 }∗ ∗ g 12 12 12 ∗ g Z (000) 1 1 0 129.108 P B ca Classification = Z deco weak { | }∗ ∗ g g ∗ Z (000) 1 1 1 029.109 P C ca Classification = Z deco weak { | 12 12 }∗ 12 12 12 ∗ g g ∗ Z (000) 1 1 1 029.110 P I ca Classification = Z deco weak { | 12 12 12 }∗ 12 12 ∗ g 12 12 ∗ g Z (000) 1 1 0 130.111 P nc Z deco weak 2 g 12 12 g 12 12 Z (000) 0 1 130.113 P n (cid:48) c (cid:48) Classification = Z × Z deco weak 2 ∗ g 12 12 g 12 12 ∗ Z (000) 1 1 0 Z (0¯20) 0 1 ¯130.114 P nc (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ g 12 12 ∗ g 12 12 Z (000) 1 0 1 Z (100) 0 0 030.115 P n (cid:48) c (cid:48) Z deco weak 2 g 12 12 ∗ g 12 12 ∗ Z (002) 0 1 130.116 P a nc Z deco weak { | }∗ g 12 12 g 12 12 Z (000) 1 0 1 130.118 P c nc Z deco weak { | }∗ g ∗ g ∗ Z (000) 1 0 0 030.119 P A nc Z deco weak { | 12 12 }∗ m ∗ m ∗ Z (000) 1 0 0 030.120 P B nc Z deco weak { | }∗ g 12 12 g 12 12 Z (000) 1 0 1 130.121 P C nc Z deco weak { | 12 12 }∗ 12 12 ∗ g 12 12 g ∗ Z (000) 1 1 1 030.122 P I nc Z deco weak { | 12 12 12 }∗ g ∗ g ∗ Z (000) 1 0 0 031.123 P mn Classification = Z M deco weak 2 g Cm Z (000) 0 1 1¯100M(100; 0) (000) 0 0 2¯20031.125 P m (cid:48) n (cid:48) Classification = Z × Z deco weak 2 ∗ g m ∗ Z (000) 1 1 0 Z (010) 0 0 031.126 P mn (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯100 Z (¯200) 0 ¯1 2011M(100; 0) (000) 0 0 2¯20031.127 P m (cid:48) n (cid:48) Classification = Z deco weak 2 g ∗ m ∗ Z (002) 1 1 031.128 P a mn Classification = Z M deco weak { | }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯100M(100; 0) (000) 0 0 0 2¯20031.129 P b mn Classification = Z M deco weak { | }∗ g Cm Z (000) 1 0 1 1¯100M(100; 0) (000) 0 0 0 2¯20031.130 P c mn Classification = Z M deco weak { | }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯100M(100; ) (000) 0 0 0 2¯20031.131 P A mn Classification = Z M deco weak { | 12 12 }∗ 12 12 ∗ g 12 12 ∗ Cm Z (000) 1 1 0 1¯100M(100; ) (000) 0 0 0 2¯20031.132 P B mn Classification = Z M deco weak { | }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯100M(100; ) (000) 0 0 0 2¯20031.133 P C mn Classification = Z M deco weak { | 12 12 }∗ 12 12 ∗ g 12 12 ∗ Cm Z (000) 1 1 0 1¯100M(100; ) (000) 0 0 0 2¯20031.134 P I mn Classification = Z M deco weak { | 12 12 12 }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯100M(100; 0) (000) 0 0 0 2¯20032.135 P ba Z deco weak 2 g 12 12 g 12 12 Z (000) 0 1 132.137 P b (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g 12 12 g 12 12 ∗ Z (000) 1 1 0 Z (0¯20) 1 0 ¯132.139 P c ba Z deco weak { | }∗ g 12 12 g 12 12 Z (000) 1 0 1 132.140 P b ba Z deco weak { | }∗ ∗ g 12 12 g ∗ Z (000) 1 1 1 032.141 P C ba Z deco weak { | 12 12 }∗ m ∗ m ∗ Z (000) 1 0 0 032.142 P A ba Z deco weak { | 12 12 }∗ g ∗ g ∗ Z (000) 1 0 0 032.143 P I ba Z deco weak { | 12 12 12 }∗ g ∗ g ∗ Z (000) 1 0 0 033.144 P na Classification = Z deco weak 2 g 12 12 g 12 12 12 Z (000) 0 1 133.146 P n (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g 12 12 g 12 12 12 ∗ Z (000) 1 1 0 Z (020) 1 0 133.147 P na (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ g 12 12 ∗ g 12 12 12 Z (000) 1 0 1 Z (¯200) 1 ¯1 033.148 P n (cid:48) a (cid:48) Classification = Z deco weak 2 g 12 12 ∗ g 12 12 12 ∗ Z (002) 1 0 133.149 P a na Classification = Z deco weak { | }∗ ∗ g 12 12 12 g ∗ Z (000) 1 1 1 033.150 P b na Classification = Z deco weak { | }∗ g ∗ g ∗ Z (000) 1 0 0 033.151 P c na Classification = Z deco weak { | }∗ ∗ g 12 12 g 12 12 ∗ Z (000) 1 1 1 033.152 P A na Classification = Z deco weak { | 12 12 }∗ ∗ g 12 12 g ∗ Z (000) 1 1 1 033.153 P B na Classification = Z deco weak { | }∗ ∗ g 12 12 12 g 12 12 ∗ Z (000) 1 1 1 033.154 P C na Classification = Z deco weak { | 12 12 }∗ g ∗ m ∗ Z (000) 1 0 0 033.155 P I na Classification = Z deco weak { | 12 12 12 }∗ 12 12 ∗ g 12 12 m ∗ Z (000) 1 1 1 034.156 P nn Z deco weak 2 g 12 12 12 g 12 12 12 Z (000) 0 1 134.158 P n (cid:48) n (cid:48) Classification = Z × Z deco weak 2 ∗ g 12 12 12 g 12 12 12 ∗ Z (000) 1 1 0 Z (0¯20) 0 1 ¯134.159 P n (cid:48) n (cid:48) Z deco weak 2 g 12 12 12 ∗ g 12 12 12 ∗ Z (002) 0 1 134.160 P a nn Z deco weak { | }∗ ∗ g 12 12 ∗ g 12 12 12 Z (000) 1 1 0 134.161 P c nn Z deco weak { | }∗ g 12 12 ∗ g 12 12 ∗ Z (000) 1 0 0 034.162 P A nn Z deco weak { | 12 12 }∗ g ∗ g ∗ Z (000) 1 0 0 034.164 P I nn Z deco weak { | 12 12 12 }∗ m ∗ m ∗ Z (000) 1 0 0 035.165 Cmm Z M deco weak 2 Cm Cm Z (000) 0 1¯1 1¯1M(010; 0) (000) 0 2¯2 00M(100; 0) (000) 0 00 2¯235.167 Cm (cid:48) m (cid:48) Classification = Z M × Z deco weak 2 ∗ m ∗ Cm Z (000) 1 0 1¯1 Z (¯1¯10) 1 0 20M(010; 0) (000) 0 0 2¯235.169 C c mm Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(010; 0) (000) 0 0 2¯2 00M(100; 0) (000) 0 0 00 2¯235.170 C a mm Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1 M(100; 0) (000) 0 0 2¯2 00M(010; 0) (000) 0 0 00 2¯235.171 C A mm Z M deco weak { | 12 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(010; 0) (000) 0 0 2¯2 00M(100; ) (000) 0 0 00 2¯236.172 Cmc Classification = Z M deco weak 2 g Cm Z (000) 0 1 1¯1M(100; ) (000) 0 0 2¯236.174 Cm (cid:48) c (cid:48) Classification = Z × Z deco weak 2 ∗ g m ∗ Z (000) 1 1 0 Z (¯1¯10) 0 0 036.175 Cmc (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯1 Z (¯110) 1 0 20M(100; ) (000) 0 0 2¯236.176 Cm (cid:48) c (cid:48) Classification = Z deco weak 2 g ∗ m ∗ Z (00¯2) ¯1 ¯1 036.177 C c mc Classification = Z M deco weak { | }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯1M(100; 0) (000) 0 0 0 2¯236.178 C a mc Classification = Z M deco weak { | 12 12 }∗ g Cm Z (000) 1 0 1 1¯1M(100; 0) (000) 0 0 0 2¯236.179 C A mc Classification = Z M deco weak { | 12 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(010; ) (000) 0 0 2¯2 00M(100; ) (000) 0 0 00 2¯237.180 Ccc Z deco weak 2 g g Z (000) 0 1 137.182 Cc (cid:48) c (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (¯1¯10) 0 0 037.184 C c cc Z deco weak { | }∗ m ∗ m ∗ Z (000) 1 0 0 037.186 C A cc Z M deco weak { | 12 12 12 }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯1M(010; ) (000) 0 0 0 2¯238.187 Amm Z M deco weak 2 Cm Cm Z (000) 0 1¯100 1¯1M(100; 0) (000) 0 1¯11¯1 00M(100; ) (000) 0 1¯1¯11 00 M(010; ) (000) 0 0000 2¯238.189 Am (cid:48) m (cid:48) Classification = Z M × Z deco weak 2 ∗ m ∗ Cm Z (000) 1 0 1¯1 Z (01¯1) 1 0 20M(010; ) (000) 0 0 2¯238.190 Amm (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ m ∗ Cm Z (000) 1 0 1¯100 Z (100) 0 0 1001M(100; 0) (000) 0 0 1¯11¯1M(100; ) (000) 0 0 1¯1¯1138.191 Am (cid:48) m (cid:48) Z deco weak 2 m ∗ m ∗ Z (011) 0 0 038.192 A a mm Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯100 1¯1M(100; 0) (000) 0 0 2¯200 00M(010; ) (000) 0 0 0000 2¯238.193 A b mm Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯100 1¯1M(100; 0) (000) 0 0 1¯11¯1 00M(100; ) (000) 0 0 1¯1¯11 00M(010; 0) (000) 0 0 0000 2¯238.194 A B mm Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯1 1¯100M(010; 0) (000) 0 0 2¯2 0000M(100; ) (000) 0 0 00 2¯20039.195 Abm Z M deco weak 2 g Cm Z (000) 0 1 1¯1M(010; ) (000) 0 0 2¯239.197 Ab (cid:48) m (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯1 Z (01¯1) 0 0 20M(010; ) (000) 0 0 2¯239.198 Abm (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ g ∗ g Z (000) 1 0 1 Z (100) 0 0 039.200 A a bm Z M deco weak { | }∗ g Cm Z (000) 1 0 1 1¯1M(010; ) (000) 0 0 0 2¯239.201 A b bm Z M deco weak { | 12 12 }∗ m ∗ Cm Z (000) 1 0 0 1¯1M(010; ) (000) 0 0 0 2¯239.202 A B bm Z deco weak { | }∗ g ∗ g ∗ Z (000) 1 0 0 0 Ama Z M deco weak 2 g Cm Z (000) 0 1 1¯100M(100; ) (000) 0 0 2¯20040.205 Am (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (01¯1) 1 1 040.206 Ama (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯100 Z (¯200) 0 ¯1 2011M(100; ) (000) 0 0 2¯20040.207 Am (cid:48) a (cid:48) Z deco weak 2 g ∗ g ∗ Z (0¯1¯1) 0 0 040.208 A a ma Z M deco weak { | }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯100M(100; ) (000) 0 0 0 2¯20040.209 A b ma Z M deco weak { | 12 12 }∗ g Cm Z (000) 1 0 1 1¯100M(100; ) (000) 0 0 0 2¯20040.210 A B ma Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯1 1¯100M(010; ) (000) 0 0 2¯2 0000M(100; ) (000) 0 0 00 2¯20041.211 Aba Z deco weak 2 g g Z (000) 0 1 141.213 Ab (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (0¯11) 0 0 041.214 Aba (cid:48) (cid:48) Classification = Z × Z deco weak 2 ∗ g ∗ g Z (000) 1 0 1 Z (¯200) 0 ¯1 141.216 A a ba Z deco weak { | }∗ g ∗ g ∗ Z (000) 1 0 0 041.217 A b ba Z deco weak { | 12 12 }∗ g ∗ g ∗ Z (000) 1 0 0 041.218 A B ba Z deco weak { | }∗ m ∗ m ∗ Z (000) 1 0 0 042.219 F mm Z M deco weak 2 Cm Cm Z (000) 0 1¯1 1¯1M(010; 0) (000) 0 2¯2 00M(100; ) (000) 0 00 2¯2 F m (cid:48) m (cid:48) Classification = Z M × Z deco weak 2 ∗ m ∗ Cm Z (000) 1 0 1¯1 Z (101) 1 0 20M(010; 0) (000) 0 0 2¯242.223 F S mm Z M deco weak { | 12 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(100; 0) (000) 0 0 2¯2 00M(010; 0) (000) 0 0 00 2¯243.224 F dd Z deco weak 2 g 14 14 14 g 14 14 14 Z (000) 0 1 143.226 F d (cid:48) d (cid:48) Classification = Z × Z deco weak 2 ∗ g 14 14 14 g 14 14 14 ∗ Z (000) 1 1 0 Z (¯20¯2) 0 1 ¯143.227 F d (cid:48) d (cid:48) Z deco weak 2 g 14 14 14 ∗ g 14 14 14 ∗ Z (220) 0 1 143.228 F S dd Z deco weak { | 12 12 12 }∗ g 14 14 14 g 14 14 14 Z (000) 1 0 1 144.229 Imm Z M deco weak 2 Cm Cm Z (000) 0 1¯1 1¯1M(010; 0) (000) 0 2¯2 00M(100; 0) (000) 0 00 2¯244.231 Im (cid:48) m (cid:48) Classification = Z M × Z deco weak 2 ∗ m ∗ Cm Z (000) 1 0 1¯1 Z (¯11¯1) 1 0 20M(010; 0) (000) 0 0 2¯244.232 Im (cid:48) m (cid:48) Z deco weak 2 m ∗ m ∗ Z (11¯1) 0 0 044.233 I c mm Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(010; 0) (000) 0 0 2¯2 00M(100; 0) (000) 0 0 00 2¯244.234 I a mm Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(100; 0) (000) 0 0 2¯2 00M(010; ) (000) 0 0 00 2¯245.235 Iba Z deco weak 2 g g Z (000) 0 1 145.237 Ib (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (1¯11) 0 0 045.239 I c ba Z deco weak { | 12 12 }∗ m ∗ m ∗ Z (000) 1 0 0 045.240 I a ba Z deco weak { | 12 12 }∗ g g Z (000) 1 0 1 146.241 Ima Z M deco weak 2 g Cm Z (000) 0 1 1¯1M(100; ) (000) 0 0 2¯246.243 Im (cid:48) a (cid:48) Classification = Z × Z deco weak 2 ∗ g g ∗ Z (000) 1 1 0 Z (¯11¯1) 1 1 046.244 Ima (cid:48) (cid:48) Classification = Z M × Z deco weak 2 ∗ g ∗ Cm Z (000) 1 0 1¯1 Z (¯111) 0 1 20M(100; ) (000) 0 0 2¯246.245 Im (cid:48) a (cid:48) Z deco weak 2 g ∗ g ∗ Z (¯1¯11) 0 0 046.246 I c ma Z M deco weak { | 12 12 }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯1M(100; ) (000) 0 0 0 2¯246.247 I a ma Z M deco weak { | 12 12 }∗ m ∗ Cm Z (000) 1 0 0 1¯1M(100; ) (000) 0 0 0 2¯246.248 I b ma Z M deco weak { | }∗ g ∗ Cm Z (000) 1 0 0 1¯1M(100; ) (000) 0 0 0 2¯247.251 P m (cid:48) mm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Cm Z (000) 1 1 0 0 1¯100 1¯100M(010; 0) (000) 0 0 0 0 1¯11¯1 0000M(010; ) (000) 0 0 0 0 1¯1¯11 0000M(001; 0) (000) 0 0 0 0 0000 1¯11¯1M(001; ) (000) 0 0 0 0 0000 1¯1¯1147.253 P m (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ P n (cid:48) nn Classification = Z deco weak 2 12 12 ∗ ∗ i ∗ 12 12 Z (000) 1 1 0 048.261 P n (cid:48) n (cid:48) n (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ 12 12 P c (cid:48) cm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯100M(001; 0) (000) 0 0 0 0 2¯20049.268 P ccm (cid:48) Classification = Z deco weak 2 ∗ i ∗ ∗ Z (000) 0 1 0 149.271 P c (cid:48) c (cid:48) m (cid:48) Classification = N/A deco weak 2 i ∗ P b (cid:48) an Classification = Z deco weak 2 12 12 ∗ ∗ i ∗ Z (000) 1 1 0 050.280 P ban (cid:48) Classification = Z deco weak 2 12 12 ∗ i ∗ ∗ Z (000) 0 1 0 150.283 P b (cid:48) a (cid:48) n (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ P m (cid:48) ma Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯100M(010; 0) (000) 0 0 0 0 1¯11¯1M(010; ) (000) 0 0 0 0 1¯1¯1151.292 P mm (cid:48) a Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Z (000) 1 0 0 1 1¯100M(100; ) (000) 0 0 0 0 2¯20051.293 P mma (cid:48) Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Cm Z (000) 0 1 0 1 1¯100 1¯100M(100; ) (000) 0 0 0 0 2¯200 0000M(010; 0) (000) 0 0 0 0 0000 1¯11¯1M(010; ) (000) 0 0 0 0 0000 1¯1¯1151.297 P m (cid:48) m (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ P n (cid:48) na Classification = Z deco weak 2 ∗ 12 12 12 ∗ i ∗ 12 12 Z (000) 1 1 0 052.308 P nn (cid:48) a Classification = Z deco weak 2 ∗ 12 12 12 i ∗ 12 12 ∗ Z (000) 1 0 0 152.309 P nna (cid:48) Classification = Z deco weak 2 12 12 12 ∗ i ∗ 12 12 ∗ Z (000) 0 1 0 152.313 P n (cid:48) n (cid:48) a (cid:48) Classification = N/Adeco weak 2 12 12 12 i ∗ 12 12 P m (cid:48) na Classification = Z deco weak 2 ∗ ∗ i ∗ Z (000) 1 1 0 053.324 P mn (cid:48) a Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Z (000) 1 0 0 1 1¯100M(100; 0) (000) 0 0 0 0 2¯20053.325 P mna (cid:48) Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Z (000) 0 1 0 1 1¯100M(100; 0) (000) 0 0 0 0 2¯20053.329 P m (cid:48) n (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ P c (cid:48) ca Classification = Z deco weak 2 ∗ ∗ i ∗ Z (000) 1 1 0 054.340 P cc (cid:48) a Classification = Z deco weak 2 ∗ i ∗ ∗ Z (000) 1 0 0 154.341 P cca (cid:48) Classification = Z deco weak 2 ∗ i ∗ ∗ Z (000) 0 1 0 154.345 P c (cid:48) c (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ P b (cid:48) am Classification = Z M deco weak 2 ∗ 12 12 ∗ i ∗ 12 12 Cm Z (000) 1 1 0 0 1¯100M(001; 0) (000) 0 0 0 0 1¯11¯1M(001; ) (000) 0 0 0 0 1¯1¯1155.356 P bam (cid:48) Classification = Z deco weak 2 12 12 ∗ i ∗ 12 12 ∗ Z (000) 0 1 0 155.359 P b (cid:48) a (cid:48) m (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ 12 12 P c (cid:48) cn Classification = Z deco weak 2 12 12 ∗ 12 12 ∗ i ∗ Z (000) 1 1 0 056.368 P ccn (cid:48) Classification = Z deco weak 2 12 12 12 12 ∗ i ∗ ∗ Z (000) 0 1 0 156.371 P c (cid:48) c (cid:48) n (cid:48) Classification = N/Adeco weak 2 12 12 12 12 i ∗ P b (cid:48) cm Classification = Z M deco weak 2 ∗ 12 12 ∗ i ∗ Cm Z (000) 1 1 0 0 1¯100M(001; ) (000) 0 0 0 0 2¯20057.380 P bc (cid:48) m Classification = Z M deco weak 2 ∗ 12 12 i ∗ ∗ Cm Z (000) 1 0 0 1 1¯100M(001; ) (000) 0 0 0 0 2¯20057.381 P bcm (cid:48) Classification = Z deco weak 2 12 12 ∗ i ∗ ∗ Z (000) 0 1 0 157.385 P b (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ P n (cid:48) nm Classification = Z M deco weak 2 ∗ 12 12 12 ∗ i ∗ 12 12 12 Cm Z (000) 1 1 0 0 1¯100M(001; 0) (000) 0 0 0 0 2¯20058.396 P nnm (cid:48) Classification = Z deco weak 2 12 12 12 ∗ i ∗ 12 12 12 ∗ Z (000) 0 1 0 158.399 P n (cid:48) n (cid:48) m (cid:48) Classification = N/Adeco weak 2 12 12 12 i ∗ 12 12 12 P m (cid:48) mn Classification = Z M deco weak 2 12 12 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯100 M(010; ) (000) 0 0 0 0 2¯20059.408 P mmn (cid:48) Classification = Z M deco weak 2 12 12 ∗ i ∗ ∗ Cm Cm Z (000) 0 1 0 1 1¯100 1¯100M(100; ) (000) 0 0 0 0 2¯200 0000M(010; ) (000) 0 0 0 0 0000 2¯20059.411 P m (cid:48) m (cid:48) n (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ P b (cid:48) cn Classification = Z deco weak 2 12 12 12 ∗ ∗ i ∗ 12 12 Z (000) 1 1 0 060.420 P bc (cid:48) n Classification = Z deco weak 2 12 12 12 ∗ i ∗ 12 12 ∗ Z (000) 1 0 0 160.421 P bcn (cid:48) Classification = Z deco weak 2 12 12 12 ∗ i ∗ 12 12 ∗ Z (000) 0 1 0 160.425 P b (cid:48) c (cid:48) n (cid:48) Classification = N/Adeco weak 2 12 12 12 i ∗ 12 12 P b (cid:48) ca Classification = Z deco weak 2 ∗ 12 12 ∗ i ∗ 12 12 Z (000) 1 1 0 061.437 P b (cid:48) c (cid:48) a (cid:48) Classification = N/Adeco weak 2 12 12 i ∗ 12 12 P n (cid:48) ma Classification = Z M deco weak 2 ∗ ∗ i ∗ 12 12 12 Cm Z (000) 1 1 0 0 1¯100M(010; ) (000) 0 0 0 0 2¯20062.444 P nm (cid:48) a Classification = Z deco weak 2 ∗ i ∗ 12 12 12 ∗ Z (000) 1 0 0 162.445 P nma (cid:48) Classification = Z M deco weak 2 ∗ i ∗ 12 12 12 ∗ Cm Z (000) 0 1 0 1 1¯100M(010; ) (000) 0 0 0 0 2¯20062.449 P n (cid:48) m (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ 12 12 12 Cm (cid:48) cm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯100M(001; ) (000) 0 0 0 0 2¯20063.460 Cmc (cid:48) m Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Cm Z (000) 1 0 0 1 1¯100 1¯1M(001; ) (000) 0 0 0 0 2¯200 00M(100; 0) (000) 0 0 0 0 0000 2¯263.461 Cmcm (cid:48) Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Z (000) 0 1 0 1 1¯1M(100; 0) (000) 0 0 0 0 2¯263.465 Cm (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ Cm (cid:48) ca Classification = Z deco weak 2 12 12 ∗ 12 12 ∗ i ∗ Z (000) 1 1 0 064.472 Cmc (cid:48) a Classification = Z M deco weak 2 12 12 ∗ 12 12 i ∗ ∗ Cm Z (000) 1 0 0 1 1¯1M(100; 0) (000) 0 0 0 0 2¯264.473 Cmca (cid:48) Classification = Z M deco weak 2 12 12 12 12 ∗ i ∗ ∗ Cm Z (000) 0 1 0 1 1¯1M(100; 0) (000) 0 0 0 0 2¯264.477 Cm (cid:48) c (cid:48) a (cid:48) Classification = N/Adeco weak 2 12 12 12 12 i ∗ Cm (cid:48) mm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Cm Z (000) 1 1 0 0 1¯1 1¯100M(010; 0) (000) 0 0 0 0 2¯2 0000M(001; 0) (000) 0 0 0 0 00 1¯11¯1M(001; ) (000) 0 0 0 0 00 1¯1¯1165.484 Cmmm (cid:48) Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Cm Z (000) 0 1 0 1 1¯1 1¯1M(010; 0) (000) 0 0 0 0 2¯2 00M(100; 0) (000) 0 0 0 0 00 2¯265.487 Cm (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ Cc (cid:48) cm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯100M(001; 0) (000) 0 0 0 0 2¯20066.494 Cccm (cid:48) Classification = Z deco weak 2 ∗ i ∗ ∗ Z (000) 0 1 0 166.497 Cc (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ Cm (cid:48) ma Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯1M(010; ) (000) 0 0 0 0 2¯267.504 Cmma (cid:48) Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Cm Z (000) 0 1 0 1 1¯1 1¯1M(010; ) (000) 0 0 0 0 2¯2 00M(100; 0) (000) 0 0 0 0 00 2¯267.507 Cm (cid:48) m (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ Cc (cid:48) ca Classification = Z deco weak 2 ∗ ∗ i ∗ 12 12 Z (000) 1 1 0 068.514 Ccca (cid:48) Classification = Z deco weak 2 ∗ i ∗ 12 12 ∗ Z (000) 0 1 0 168.517 Cc (cid:48) c (cid:48) a (cid:48) Classification = N/A deco weak 2 i ∗ 12 12 F m (cid:48) mm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Cm Z (000) 1 1 0 0 1¯1 1¯1M(001; 0) (000) 0 0 0 0 2¯2 00M(010; 0) (000) 0 0 0 0 00 2¯269.525 F m (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ F d (cid:48) dd Classification = Z deco weak 2 14 14 ∗ ∗ i ∗ 14 14 Z (000) 1 1 0 070.531 F d (cid:48) d (cid:48) d (cid:48) Classification = N/Adeco weak 2 14 14 i ∗ 14 14 Im (cid:48) mm Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Cm Z (000) 1 1 0 0 1¯1 1¯1M(001; 0) (000) 0 0 0 0 2¯2 00M(010; 0) (000) 0 0 0 0 00 2¯271.537 Im (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ Ib (cid:48) am Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯1M(001; 0) (000) 0 0 0 0 2¯272.542 Ibam (cid:48) Classification = Z deco weak 2 ∗ i ∗ ∗ Z (000) 0 1 0 172.545 Ib (cid:48) a (cid:48) m (cid:48) Classification = N/Adeco weak 2 i ∗ Ib (cid:48) ca Classification = Z deco weak 2 ∗ ∗ i ∗ Z (000) 1 1 0 073.552 Ib (cid:48) c (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ Im (cid:48) ma Classification = Z M deco weak 2 ∗ ∗ i ∗ Cm Z (000) 1 1 0 0 1¯1M(010; ) (000) 0 0 0 0 2¯274.557 Imma (cid:48) Classification = Z M deco weak 2 ∗ i ∗ ∗ Cm Cm Z (000) 0 1 0 1 1¯1 1¯1M(010; ) (000) 0 0 0 0 2¯2 00M(100; 0) (000) 0 0 0 0 00 2¯274.560 Im (cid:48) m (cid:48) a (cid:48) Classification = N/Adeco weak 2 i ∗ P (cid:48) Classification = Z deco weak 4 ∗ Z (000) 175.4 P c Z deco weak { | }∗ Z (000) 1 075.6 P I Z deco weak { | 12 12 12 }∗ Z (000) 1 076.7 P Classification = Z deco weak 4 Z (00¯4) ¯176.9 P (cid:48) Classification = Z deco weak 4 ∗ Z (000) 176.10 P c Classification = Z deco weak { | }∗ Z (000) 1 076.11 P C Classification = Z deco weak { | 12 12 }∗ Z (000) 1 076.12 P I Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 077.15 P (cid:48) Classification = Z deco weak 4 ∗ Z (000) 177.16 P c Classification = Z deco weak { | }∗ ∗ Z (000) 1 177.18 P I Classification = Z deco weak { | 12 12 12 }∗ 12 12 ∗ Z (000) 1 178.19 P Classification = Z deco weak 4 Z (00¯4) ¯378.21 P (cid:48) Classification = Z deco weak 4 ∗ Z (000) 178.22 P c Classification = Z deco weak { | }∗ ∗ Z (000) 1 178.23 P C Classification = Z deco weak { | 12 12 }∗ Z (000) 1 078.24 P I Classification = Z deco weak { | 12 12 12 }∗ 12 12 14 ∗ Z (000) 1 179.27 I (cid:48) Classification = Z deco weak 4 ∗ Z (000) 179.28 I c Z deco weak { | 12 12 }∗ Z (000) 1 080.29 I Classification = Z deco weak 4 12 14 Z (22¯2) 180.31 I (cid:48) Classification = Z deco weak 4 12 14 ∗ Z (000) 180.32 I c Classification = Z deco weak { | 12 12 }∗ 12 14 Z (000) 1 081.35 P ¯4 (cid:48) Classification = N/Adeco weak ¯4 ∗ ¯4 ∗ I ¯4 (cid:48) Classification = N/Adeco weak ¯4 ∗ ¯4 ∗ P /m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ ¯4 ∗ P /m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ ¯4 ∗ P /n (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ ¯4 ∗ P /n (cid:48) Classification = N/Adeco weak 4 12 12 i ∗ ¯4 12 12 ∗ ¯4 ∗ I /m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ ¯4 ∗ I /a (cid:48) Classification = N/Adeco weak 4 14 14 14 i ∗ ¯4 14 14 14 ∗ ¯4 14 14 14 ∗ P 422 Classification = N/Adeco weak 4 P (cid:48) (cid:48) Classification = Z deco weak 4 ∗ ∗ Z (000) 1 0 189.91 P (cid:48) (cid:48) Z deco weak 4 ∗ ∗ Z (000) 1 1 089.92 P c 422 Classification = Z deco weak { | }∗ Z (000) 1 0 0 089.93 P C 422 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 089.94 P I 422 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0 090.95 P 12 12 12 12 P (cid:48) (cid:48) Classification = Z deco weak 4 12 12 ∗ 12 12 ∗ Z (000) 1 0 190.99 P (cid:48) (cid:48) Z deco weak 4 12 12 ∗ 12 12 ∗ Z (000) 1 1 090.100 P c Z deco weak { | }∗ 12 12 12 12 Z (000) 1 0 0 090.101 P C Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 1 0 190.102 P I Z deco weak { | 12 12 12 }∗ 12 12 ∗ 12 12 12 ∗ Z (000) 1 0 1 1 P 22 Classification = N/Adeco weak 4 P (cid:48) (cid:48) Classification = Z deco weak 4 ∗ ∗ Z (000) 1 0 191.106 P (cid:48) (cid:48) Classification = Z × Z deco weak 4 ∗ ∗ Z (000) 0 1 1 Z (00¯4) ¯1 1 091.107 P (cid:48) (cid:48) Z deco weak 4 ∗ ∗ Z (000) 1 1 091.108 P c 22 Classification = Z deco weak { | }∗ Z (000) 1 0 0 091.109 P C 22 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 091.110 P I 22 Classification = Z deco weak { | 12 12 12 }∗ 12 12 34 ∗ Z (000) 1 0 0 192.111 P 12 12 14 12 12 34 P (cid:48) (cid:48) Classification = Z deco weak 4 12 12 14 ∗ 12 12 34 ∗ Z (000) 1 0 192.114 P (cid:48) (cid:48) Classification = Z × Z deco weak 4 12 12 14 12 12 34 ∗ ∗ Z (000) 0 1 1 Z (00¯4) ¯1 1 092.115 P (cid:48) (cid:48) Z deco weak 4 12 12 14 ∗ 12 12 34 ∗ Z (000) 1 1 092.116 P c Z deco weak { | }∗ 12 12 14 12 12 14 ∗ Z (000) 1 0 0 192.117 P C Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 1 0 192.118 P I Z deco weak { | 12 12 12 }∗ 12 12 14 ∗ Z (000) 1 0 1 093.119 P 22 Classification = N/Adeco weak 4 P (cid:48) (cid:48) Classification = Z deco weak 4 ∗ ∗ Z (000) 1 0 193.123 P (cid:48) (cid:48) Z deco weak 4 ∗ ∗ Z (000) 1 1 093.124 P c 22 Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 0 193.125 P C 22 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 093.126 P I 22 Classification = Z deco weak { | 12 12 12 }∗ 12 12 ∗ 12 12 ∗ Z (000) 1 1 0 194.127 P 12 12 12 12 12 12 P (cid:48) (cid:48) Classification = Z deco weak 4 12 12 12 ∗ 12 12 12 ∗ Z (000) 1 0 194.131 P (cid:48) (cid:48) Z deco weak 4 12 12 12 ∗ 12 12 12 ∗ Z (000) 1 1 094.132 P c Z deco weak { | }∗ 12 12 ∗ 12 12 ∗ Z (000) 1 1 1 094.133 P C Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 1 0 194.134 P I Z deco weak { | 12 12 12 }∗ ∗ ∗ Z (000) 1 1 1 095.135 P 22 Classification = N/Adeco weak 4 P (cid:48) (cid:48) Classification = Z deco weak 4 ∗ ∗ Z (000) 1 0 195.138 P (cid:48) (cid:48) Classification = Z × Z deco weak 4 ∗ ∗ Z (000) 0 1 1 Z (004) 3 1 095.139 P (cid:48) (cid:48) Z deco weak 4 ∗ ∗ Z (000) 1 1 095.140 P c 22 Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 0 195.141 P C 22 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 095.142 P I 22 Classification = Z deco weak { | 12 12 12 }∗ 12 12 14 ∗ Z (000) 1 1 0 096.143 P 12 12 34 12 12 14 P (cid:48) (cid:48) Classification = Z deco weak 4 12 12 34 ∗ 12 12 14 ∗ Z (000) 1 0 196.146 P (cid:48) (cid:48) Classification = Z × Z deco weak 4 12 12 34 12 12 14 ∗ ∗ Z (000) 0 1 1 Z (004) 3 1 096.147 P (cid:48) (cid:48) Z deco weak 4 12 12 34 ∗ 12 12 14 ∗ Z (000) 1 1 096.148 P c Z deco weak { | }∗ 12 12 14 ∗ 12 12 14 ∗ ∗ Z (000) 1 1 1 196.149 P C Z deco weak { | 12 12 }∗ ∗ ∗ Z (000) 1 1 0 196.150 P I Z deco weak { | 12 12 12 }∗ ∗ ∗ 12 12 12 ∗ Z (000) 1 1 1 197.151 I 422 Classification = N/Adeco weak 4 I (cid:48) (cid:48) Classification = Z deco weak 4 ∗ ∗ Z (000) 1 0 197.155 I (cid:48) (cid:48) Z deco weak 4 ∗ ∗ Z (000) 1 1 097.156 I c 422 Classification = Z deco weak { | 12 12 }∗ Z (000) 1 0 0 098.157 I 22 Classification = N/Adeco weak 4 12 14 12 14 I (cid:48) (cid:48) Classification = Z deco weak 4 12 14 ∗ 12 14 ∗ Z (000) 1 0 198.160 I (cid:48) (cid:48) Classification = Z × Z deco weak 4 12 14 12 14 ∗ ∗ Z (000) 0 1 1 Z (22¯2) 1 0 198.161 I (cid:48) (cid:48) Z deco weak 4 12 14 ∗ 12 14 ∗ Z (000) 1 1 098.162 I c 22 Classification = Z deco weak { | 12 12 }∗ 12 14 12 14 Z (000) 1 0 0 099.163 P mm Classification = Z M deco weak 4 Cm Cm Z (000) 0 1¯100 1¯1M(100; 0) (000) 0 1¯11¯1 00M(100; ) (000) 0 1¯1¯11 00M(1¯10; 0) (000) 0 0000 2¯299.165 P (cid:48) m (cid:48) m Classification = Z M deco weak 4 ∗ m ∗ Cm Z (000) 1 0 1¯1M(1¯10; 0) (000) 0 0 2¯299.166 P (cid:48) mm (cid:48) Classification = Z M deco weak 4 ∗ m ∗ Cm Z (000) 1 0 1¯100M(100; 0) (000) 0 0 1¯11¯1 M(100; ) (000) 0 0 1¯1¯1199.168 P c mm Classification = Z M deco weak { | }∗ Cm Cm Z (000) 1 0 1¯100 1¯1M(100; 0) (000) 0 0 1¯11¯1 00M(100; ) (000) 0 0 1¯1¯11 00M(1¯10; 0) (000) 0 0 0000 2¯299.169 P C mm Classification = Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯100M(110; 0) (000) 0 0 2¯2 0000M(100; 0) (000) 0 0 00 2¯20099.170 P I mm Classification = Z M deco weak { | 12 12 12 }∗ Cm Cm Z (000) 1 0 1¯100 1¯1M(010; 0) (000) 0 0 2¯200 00M(1¯10; 0) (000) 0 0 0000 2¯2100.171 P bm Classification = Z M deco weak 4 g 12 12 Cm Z (000) 0 1 1¯1M(1¯10; ) (000) 0 0 2¯2100.173 P (cid:48) b (cid:48) m Classification = Z M deco weak 4 ∗ g 12 12 ∗ Cm Z (000) 1 0 1¯1M(1¯10; ) (000) 0 0 2¯2100.174 P (cid:48) bm (cid:48) Classification = Z deco weak 4 ∗ g 12 12 g 12 12 ∗ Z (000) 1 1 0100.176 P c bm Classification = Z M deco weak { | }∗ g 12 12 Cm Z (000) 1 0 1 1¯1M(1¯10; ) (000) 0 0 0 2¯2100.177 P C bm Classification = Z M deco weak { | 12 12 }∗ 12 12 ∗ m ∗ Cm Z (000) 1 1 0 1¯1M(110; ) (000) 0 0 0 2¯2100.178 P I bm Classification = Z M deco weak { | 12 12 12 }∗ g ∗ Cm Z (000) 1 0 0 1¯1M(110; ) (000) 0 0 0 2¯2101.179 P cm Classification = Z M deco weak 4 g Cm Z (000) 0 1 1¯1M(110; 0) (000) 0 0 2¯2101.181 P (cid:48) c (cid:48) m Classification = Z M deco weak 4 ∗ g ∗ Cm Z (000) 1 0 1¯1M(110; 0) (000) 0 0 2¯2101.182 P (cid:48) cm (cid:48) Classification = Z deco weak 4 ∗ g m ∗ Z (000) 1 1 0101.184 P c cm Classification = Z M deco weak { | }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯1 M(1¯10; 0) (000) 0 0 0 2¯2101.185 P C cm Classification = Z M deco weak { | 12 12 }∗ g Cm Z (000) 1 0 1 1¯1M(110; 0) (000) 0 0 0 2¯2101.186 P I cm Classification = Z M deco weak { | 12 12 12 }∗ 12 12 ∗ g Cm Z (000) 1 1 1 1¯1M(110; 0) (000) 0 0 0 2¯2102.187 P nm Classification = Z M deco weak 4 12 12 12 g 12 12 12 Cm Z (000) 0 1 1¯1M(110; 0) (000) 0 0 2¯2102.189 P (cid:48) n (cid:48) m Classification = Z M deco weak 4 12 12 12 ∗ g 12 12 12 ∗ Cm Z (000) 1 0 1¯1M(110; 0) (000) 0 0 2¯2102.190 P (cid:48) nm (cid:48) Classification = Z deco weak 4 12 12 12 ∗ g 12 12 12 m ∗ Z (000) 1 1 0102.192 P c nm Classification = Z M deco weak { | }∗ 12 12 ∗ g 12 12 ∗ Cm Z (000) 1 1 0 1¯1M(1¯10; 0) (000) 0 0 0 2¯2102.193 P C nm Classification = Z M deco weak { | 12 12 }∗ ∗ g ∗ Cm Z (000) 1 1 0 1¯1M(110; 0) (000) 0 0 0 2¯2102.194 P I nm Classification = Z M deco weak { | 12 12 12 }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯1M(1¯10; 0) (000) 0 0 0 2¯2103.195 P cc Classification = Z deco weak 4 g g Z (000) 0 1 1103.197 P (cid:48) c (cid:48) c Classification = Z deco weak 4 ∗ g ∗ g Z (000) 1 0 1103.198 P (cid:48) cc (cid:48) Classification = Z deco weak 4 ∗ g g ∗ Z (000) 1 1 0103.200 P c cc Classification = Z deco weak { | }∗ m ∗ m ∗ Z (000) 1 0 0 0103.201 P C cc Classification = Z deco weak { | 12 12 }∗ g g Z (000) 1 0 1 1103.202 P I cc Classification = Z deco weak { | 12 12 12 }∗ g g Z (000) 1 0 1 1104.203 P nc Classification = Z deco weak 4 g 12 12 12 g 12 12 12 Z (000) 0 1 1 P (cid:48) n (cid:48) c Classification = Z deco weak 4 ∗ g 12 12 12 ∗ g 12 12 12 Z (000) 1 0 1104.206 P (cid:48) nc (cid:48) Classification = Z deco weak 4 ∗ g 12 12 12 g 12 12 12 ∗ Z (000) 1 1 0104.208 P c nc Classification = Z deco weak { | }∗ g 12 12 ∗ g 12 12 ∗ Z (000) 1 0 0 0104.210 P I nc Classification = Z deco weak { | 12 12 12 }∗ m ∗ m ∗ Z (000) 1 0 0 0105.211 P mc Classification = Z M deco weak 4 g Cm Z (000) 0 1 1¯100M(100; 0) (000) 0 0 1¯11¯1M(100; ) (000) 0 0 1¯1¯11105.213 P (cid:48) m (cid:48) c Classification = Z deco weak 4 ∗ m ∗ g Z (000) 1 0 1105.214 P (cid:48) mc (cid:48) Classification = Z M deco weak 4 ∗ g ∗ Cm Z (000) 1 0 1¯100M(100; 0) (000) 0 0 1¯11¯1M(100; ) (000) 0 0 1¯1¯11105.216 P c mc Classification = Z M deco weak { | }∗ ∗ m ∗ Cm Z (000) 1 1 0 1¯100M(100; 0) (000) 0 0 0 1¯11¯1M(100; ) (000) 0 0 0 1¯1¯11105.217 P C mc Classification = Z M deco weak { | 12 12 }∗ g Cm Z (000) 1 0 1 1¯100M(010; 0) (000) 0 0 0 2¯200105.218 P I mc Classification = Z M deco weak { | 12 12 12 }∗ 12 12 ∗ g 12 12 ∗ Cm Z (000) 1 1 0 1¯100M(010; 0) (000) 0 0 0 2¯200106.219 P bc Classification = Z deco weak 4 g 12 12 g 12 12 12 Z (000) 0 1 1106.221 P (cid:48) b (cid:48) c Classification = Z deco weak 4 ∗ g 12 12 ∗ g 12 12 12 Z (000) 1 0 1106.222 P (cid:48) bc (cid:48) Classification = Z deco weak 4 ∗ g 12 12 g 12 12 12 ∗ Z (000) 1 1 0106.224 P c bc Classification = Z deco weak { | }∗ ∗ g 12 12 g 12 12 ∗ Z (000) 1 1 1 0106.225 P C bc Classification = Z deco weak { | 12 12 }∗ 12 12 12 ∗ g 12 12 12 m ∗ Z (000) 1 1 1 0 P I bc Classification = Z deco weak { | 12 12 12 }∗ 12 12 ∗ g ∗ g 12 12 12 Z (000) 1 1 0 1107.227 I mm Classification = Z M deco weak 4 Cm Cm Z (000) 0 1¯1 1¯1M(010; 0) (000) 0 2¯2 00M(1¯10; 0) (000) 0 00 2¯2107.229 I (cid:48) m (cid:48) m Classification = Z M deco weak 4 ∗ m ∗ Cm Z (000) 1 0 1¯1M(1¯10; 0) (000) 0 0 2¯2107.230 I (cid:48) mm (cid:48) Classification = Z M deco weak 4 ∗ m ∗ Cm Z (000) 1 0 1¯1M(010; 0) (000) 0 0 2¯2107.232 I c mm Classification = Z M deco weak { | 12 12 }∗ Cm Cm Z (000) 1 0 1¯1 1¯1M(110; 0) (000) 0 0 2¯2 00M(100; 0) (000) 0 0 00 2¯2108.233 I cm Classification = Z M deco weak 4 g Cm Z (000) 0 1 1¯1M(110; ) (000) 0 0 2¯2108.235 I (cid:48) c (cid:48) m Classification = Z M deco weak 4 ∗ g ∗ Cm Z (000) 1 0 1¯1M(110; ) (000) 0 0 2¯2108.236 I (cid:48) cm (cid:48) Classification = Z deco weak 4 ∗ g g ∗ Z (000) 1 1 0108.238 I c cm Classification = Z M deco weak { | 12 12 }∗ m ∗ Cm Z (000) 1 0 0 1¯1M(1¯10; ) (000) 0 0 0 2¯2109.239 I md Classification = Z M deco weak 4 12 14 g 12 14 Cm Z (000) 0 1 1¯1M(010; 0) (000) 0 0 2¯2109.241 I (cid:48) m (cid:48) d Classification = Z deco weak 4 12 14 ∗ m ∗ g 12 14 Z (000) 1 0 1109.242 I (cid:48) md (cid:48) Classification = Z M deco weak 4 12 14 ∗ g 12 14 ∗ Cm Z (000) 1 0 1¯1M(010; 0) (000) 0 0 2¯2109.243 I m (cid:48) d (cid:48) Classification = Z deco weak 4 12 14 m ∗ g 12 14 ∗ Z (22¯2) 1 0 1109.244 I c md Classification = Z M deco weak { | 12 12 }∗ 12 14 g 12 14 Cm Z (000) 1 0 1 1¯1M(010; 0) (000) 0 0 0 2¯2 I cd Classification = Z deco weak 4 12 14 g g Z (000) 0 1 1110.247 I (cid:48) c (cid:48) d Classification = Z deco weak 4 12 14 ∗ g ∗ g Z (000) 1 0 1110.248 I (cid:48) cd (cid:48) Classification = Z deco weak 4 12 14 ∗ g g ∗ Z (000) 1 1 0110.250 I c cd Classification = Z deco weak { | 12 12 }∗ 12 14 g 12 14 ∗ m ∗ Z (000) 1 0 0 0111.253 P ¯4 (cid:48) (cid:48) m Classification = Z M deco weak ¯4 ∗ ∗ Cm Z (000) 0 1 1¯1M(110; 0) (000) 0 0 2¯2111.254 P ¯4 (cid:48) m (cid:48) Classification = N/Adeco weak ¯4 ∗ m ∗ P ¯4 (cid:48) (cid:48) c Classification = Z deco weak ¯4 ∗ ∗ g Z (000) 0 1 1112.262 P ¯4 (cid:48) c (cid:48) Classification = N/Adeco weak ¯4 ∗ g ∗ P ¯4 (cid:48) (cid:48) m Classification = Z M deco weak ¯4 ∗ 12 12 ∗ Cm Z (000) 0 1 1¯1M(1¯10; ) (000) 0 0 2¯2113.270 P ¯4 (cid:48) m (cid:48) Classification = N/Adeco weak ¯4 ∗ 12 12 g 12 12 ∗ P ¯4 (cid:48) (cid:48) c Classification = Z deco weak ¯4 ∗ 12 12 12 ∗ g 12 12 12 Z (000) 0 1 1114.278 P ¯4 (cid:48) c (cid:48) Classification = N/Adeco weak ¯4 ∗ 12 12 12 g 12 12 12 ∗ P ¯4 (cid:48) m (cid:48) ∗ m ∗ P ¯4 (cid:48) m (cid:48) Classification = Z M deco weak ¯4 ∗ ∗ Cm Z (000) 0 1 1¯100M(100; 0) (000) 0 0 1¯11¯1M(100; ) (000) 0 0 1¯1¯11116.293 P ¯4 (cid:48) c (cid:48) ∗ g ∗ P ¯4 (cid:48) c (cid:48) Classification = Z deco weak ¯4 ∗ g ∗ Z (000) 0 1 1117.301 P ¯4 (cid:48) b (cid:48) ∗ g 12 12 ∗ 12 12 P ¯4 (cid:48) b (cid:48) Classification = Z deco weak ¯4 ∗ g 12 12 12 12 ∗ Z (000) 0 1 1118.309 P ¯4 (cid:48) n (cid:48) deco weak ¯4 ∗ g 12 12 12 ∗ 12 12 12 P ¯4 (cid:48) n (cid:48) Classification = Z deco weak ¯4 ∗ g 12 12 12 12 12 12 ∗ Z (000) 0 1 1119.317 I ¯4 (cid:48) m (cid:48) ∗ m ∗ I ¯4 (cid:48) m (cid:48) Classification = Z M deco weak ¯4 ∗ ∗ Cm Z (000) 0 1 1¯1M(010; 0) (000) 0 0 2¯2120.323 I ¯4 (cid:48) c (cid:48) ∗ g ∗ I ¯4 (cid:48) c (cid:48) Classification = Z deco weak ¯4 ∗ g ∗ Z (000) 0 1 1121.329 I ¯4 (cid:48) (cid:48) m Classification = Z M deco weak ¯4 ∗ ∗ Cm Z (000) 0 1 1¯1M(110; 0) (000) 0 0 2¯2121.330 I ¯4 (cid:48) m (cid:48) Classification = N/Adeco weak ¯4 ∗ m ∗ I ¯4 (cid:48) (cid:48) d Classification = Z deco weak ¯4 ∗ 12 14 ∗ g 12 14 Z (000) 0 1 1122.336 I ¯4 (cid:48) d (cid:48) Classification = N/Adeco weak ¯4 ∗ 12 14 g 12 14 ∗ P /m (cid:48) mm Classification = Z M deco weak 4 ∗ i ∗ ∗ ¯4 ∗ Cm Cm Z (000) 0 1 0 1 0 1¯100 1¯1M(100; 0) (000) 0 0 0 0 0 1¯11¯1 00M(100; ) (000) 0 0 0 0 0 1¯1¯11 00M(1¯10; 0) (000) 0 0 0 0 0 0000 2¯2123.347 P /m (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ m ∗ P /m (cid:48) cc Classification = Z deco weak 4 ∗ i ∗ ∗ ¯4 ∗ g Z (000) 0 1 0 1 0 1124.359 P /m (cid:48) c (cid:48) c (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ g ∗ P /n (cid:48) bm Classification = Z M deco weak 4 ∗ i ∗ ∗ ¯4 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(110; 0) (000) 0 0 0 0 0 2¯2125.371 P /n (cid:48) b (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ g 12 12 ∗ P /n (cid:48) nc Classification = Z deco weak 4 12 12 ∗ i ∗ ∗ ¯4 ∗ g 12 12 12 Z (000) 0 1 0 1 0 1126.383 P /n (cid:48) n (cid:48) c (cid:48) Classification = N/Adeco weak 4 12 12 i ∗ ¯4 ∗ g 12 12 12 ∗ P /m (cid:48) bm Classification = Z M deco weak 4 12 12 ∗ i ∗ 12 12 ∗ ¯4 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(1¯10; ) (000) 0 0 0 0 0 2¯2127.395 P /m (cid:48) b (cid:48) m (cid:48) Classification = N/Adeco weak 4 12 12 i ∗ 12 12 ¯4 ∗ g 12 12 ∗ P /m (cid:48) nc Classification = Z deco weak 4 12 12 12 ∗ i ∗ 12 12 12 ∗ ¯4 ∗ g 12 12 12 Z (000) 0 1 0 1 0 1128.407 P /m (cid:48) n (cid:48) c (cid:48) Classification = N/Adeco weak 4 12 12 12 i ∗ 12 12 12 ¯4 ∗ g 12 12 12 ∗ P /n (cid:48) mm Classification = Z M deco weak 4 ∗ i ∗ 12 12 ∗ ¯4 ∗ Cm Cm Z (000) 0 1 0 1 0 1¯100 1¯1M(100; ) (000) 0 0 0 0 0 2¯200 00M(1¯10; 0) (000) 0 0 0 0 0 0000 2¯2129.419 P /n (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ 12 12 ¯4 ∗ m ∗ P /n (cid:48) cc Classification = Z deco weak 4 ∗ i ∗ 12 12 12 ∗ ¯4 ∗ g Z (000) 0 1 0 1 0 1130.431 P /n (cid:48) c (cid:48) c (cid:48) Classification = N/Adeco weak 4 i ∗ 12 12 12 ¯4 ∗ g ∗ P /m (cid:48) mc Classification = Z M deco weak 4 ∗ i ∗ ∗ ¯4 ∗ g Cm Z (000) 0 1 0 1 0 1 1¯100M(100; 0) (000) 0 0 0 0 0 0 1¯11¯1M(100; ) (000) 0 0 0 0 0 0 1¯1¯11131.443 P /m (cid:48) m (cid:48) c (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ g ∗ P /m (cid:48) cm Classification = Z M deco weak 4 ∗ i ∗ ∗ ¯4 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(110; 0) (000) 0 0 0 0 0 2¯2132.455 P /m (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ m ∗ P /n (cid:48) bc Classification = Z deco weak 4 ∗ i ∗ ∗ ¯4 ∗ g 12 12 12 Z (000) 0 1 0 1 0 1133.467 P /n (cid:48) b (cid:48) c (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ g 12 12 12 ∗ P /n (cid:48) nm Classification = Z M deco weak 4 12 12 ∗ i ∗ ∗ ¯4 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(110; 0) (000) 0 0 0 0 0 2¯2134.479 P /n (cid:48) n (cid:48) m (cid:48) Classification = N/Adeco weak 4 12 12 i ∗ ¯4 ∗ g 12 12 ∗ P /m (cid:48) bc Classification = Z deco weak 4 12 12 ∗ i ∗ 12 12 12 ∗ ¯4 ∗ g 12 12 12 Z (000) 0 1 0 1 0 1135.491 P /m (cid:48) b (cid:48) c (cid:48) Classification = N/Adeco weak 4 12 12 i ∗ 12 12 12 ¯4 ∗ g 12 12 12 ∗ P /m (cid:48) nm Classification = Z M deco weak 4 12 12 12 12 12 12 ∗ i ∗ ∗ ¯4 12 12 12 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(110; 0) (000) 0 0 0 0 0 2¯2136.503 P /m (cid:48) n (cid:48) m (cid:48) Classification = N/Adeco weak 4 12 12 12 12 12 12 i ∗ ¯4 12 12 12 ∗ m ∗ P /n (cid:48) mc Classification = Z M deco weak 4 ∗ i ∗ 12 12 12 ∗ ¯4 ∗ g Cm Z (000) 0 1 0 1 0 1 1¯100M(100; ) (000) 0 0 0 0 0 0 2¯200137.515 P /n (cid:48) m (cid:48) c (cid:48) Classification = N/Adeco weak 4 i ∗ 12 12 12 ¯4 ∗ g ∗ P /n (cid:48) cm Classification = Z M deco weak 4 ∗ i ∗ 12 12 ∗ ¯4 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(110; ) (000) 0 0 0 0 0 2¯2138.527 P /n (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ 12 12 ¯4 ∗ m ∗ I /m (cid:48) mm Classification = Z M deco weak 4 ∗ i ∗ ∗ ¯4 ∗ Cm Cm Z (000) 0 1 0 1 0 1¯1 1¯1M(010; 0) (000) 0 0 0 0 0 2¯2 00M(1¯10; 0) (000) 0 0 0 0 0 00 2¯2139.539 I /m (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ m ∗ I /m (cid:48) cm Classification = Z M deco weak 4 ∗ i ∗ ∗ ¯4 ∗ Cm Z (000) 0 1 0 1 0 1¯1M(110; ) (000) 0 0 0 0 0 2¯2140.549 I /m (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 4 i ∗ ¯4 ∗ g ∗ I /a (cid:48) md Classification = Z M deco weak 4 ¯ 14 14 ∗ i ∗ ¯ 14 14 ∗ ¯4 ¯ 14 14 ∗ g 14 14 ¯ Cm Z (000) 0 1 0 1 0 1 1¯1M(010; ) (000) 0 0 0 0 0 0 2¯2141.559 I /a (cid:48) m (cid:48) d (cid:48) Classification = N/Adeco weak 4 ¯ 14 14 i ∗ ¯ 14 14 ¯4 ¯ 14 14 ∗ g 14 14 ¯ ∗ I /a (cid:48) cd Classification = Z deco weak 4 ¯ 14 14 ∗ i ∗ 14 14 14 ∗ ¯4 ¯ 14 14 ∗ g 14 14 14 Z (000) 0 1 0 1 0 1142.569 I /a (cid:48) c (cid:48) d (cid:48) Classification = N/Adeco weak 4 ¯ 14 14 i ∗ 14 14 14 ¯4 ¯ 14 14 ∗ g 14 14 14 ∗ P c Z deco weak { | }∗ Z (000) 1 0144.4 P Classification = Z deco weak 3 Z (00¯3) ¯1144.6 P c Classification = Z deco weak { | }∗ Z (000) 1 0145.7 P Classification = Z deco weak 3 Z (00¯3) ¯2145.9 P c Classification = Z deco weak { | }∗ ∗ Z (000) 1 1146.10 R Z deco weak 3 Z (111) 0146.12 R I Z deco weak { | 12 12 12 }∗ Z (000) 1 0147.15 P ¯3 (cid:48) Classification = N/Adeco weak 3 i ∗ R ¯3 (cid:48) Classification = N/Adeco weak 3 i ∗ P 312 Classification = N/Adeco weak 3 P c 312 Classification = Z deco weak { | }∗ Z (000) 1 0 0150.25 P 321 Classification = N/Adeco weak 3 P c 321 Classification = Z deco weak { | }∗ Z (000) 1 0 0151.29 P 12 Classification = N/Adeco weak 3 P (cid:48) Classification = Z × Z deco weak 3 ∗ Z (000) 0 1 Z (00¯3) ¯1 1151.32 P c 12 Classification = Z deco weak { | }∗ Z (000) 1 0 0152.33 P 21 Classification = N/Adeco weak 3 P (cid:48) Z × Z deco weak 3 ∗ Z (000) 0 1 Z (00¯3) ¯1 1152.36 P c 21 Classification = Z deco weak { | }∗ ∗ Z (000) 1 0 1153.37 P 12 Classification = N/Adeco weak 3 P (cid:48) Classification = Z × Z deco weak 3 ∗ Z (000) 0 1 Z (00¯3) ¯2 1153.40 P c 12 Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 1154.41 P 21 Classification = N/A deco weak 3 P (cid:48) Z × Z deco weak 3 ∗ Z (000) 0 1 Z (00¯3) ¯2 1154.44 P c 21 Classification = Z deco weak { | }∗ ∗ Z (000) 1 1 0155.45 R 32 Classification = N/Adeco weak 3 R (cid:48) Classification = Z × Z deco weak 3 ∗ Z (000) 0 1 Z (111) 0 0155.48 R I 32 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0156.49 P m Z M deco weak 3 Cm Z (000) 0 1¯1M(110; ) (000) 0 2¯2156.52 P c m Z M deco weak { | }∗ Cm Z (000) 1 0 1¯1M(110; ) (000) 0 0 2¯2157.53 P m Classification = Z M deco weak 3 Cm Z (000) 0 1¯1M(100; 0) (000) 0 2¯2157.56 P c m Classification = Z M deco weak { | }∗ Cm Z (000) 1 0 1¯1M(100; 0) (000) 0 0 2¯2158.57 P c Z deco weak 3 g ¯21000 Z (000) 0 1158.60 P c c Z deco weak { | }∗ m ¯210 ∗ Z (000) 1 0 0159.61 P c Classification = Z deco weak 3 g Z (000) 0 1159.64 P c c Classification = Z deco weak { | }∗ m ∗ Z (000) 1 0 0160.65 R m Classification = Z M deco weak 3 Cm Z (000) 0 1¯1M(110; ) (000) 0 2¯2160.67 R m (cid:48) Classification = Z deco weak 3 m ¯210 ∗ Z (111) 0 0160.68 R I m Classification = Z M deco weak { | 12 12 12 }∗ Cm Z (000) 1 0 1¯1 M(110; ) (000) 0 0 2¯2161.69 R c Classification = Z deco weak 3 g ¯21000 Z (000) 0 1161.71 R c (cid:48) Classification = Z deco weak 3 g ¯21000 ∗ Z (¯2¯2¯2) 0 ¯3161.72 R I c Classification = Z deco weak { | 12 12 12 }∗ m ¯210 ∗ Z (000) 1 0 0162.75 P ¯3 (cid:48) m Classification = Z M deco weak 3 ∗ i ∗ Cm Z (000) 0 1 0 1¯1M(100; 0) (000) 0 0 0 2¯2162.76 P ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ P ¯3 (cid:48) c Classification = Z deco weak 3 ∗ i ∗ Z (000) 0 1 0163.82 P ¯3 (cid:48) c (cid:48) Classification = N/Adeco weak 3 i ∗ P ¯3 (cid:48) m Z M deco weak 3 ∗ i ∗ Cm Z (000) 0 1 0 1¯1M(110; ) (000) 0 0 0 2¯2164.88 P ¯3 (cid:48) m (cid:48) i ∗ P ¯3 (cid:48) c Z deco weak 3 ∗ i ∗ Z (000) 0 1 0165.94 P ¯3 (cid:48) c (cid:48) i ∗ R ¯3 (cid:48) m Classification = Z M deco weak 3 ∗ i ∗ Cm ¯210(0) Z (000) 0 1 0 1¯1M(¯210; 0) (000) 0 0 0 2¯2166.100 R ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ R ¯3 (cid:48) c Classification = Z deco weak 3 ∗ i ∗ Z (000) 0 1 0167.106 R ¯3 (cid:48) c (cid:48) Classification = N/Adeco weak 3 i ∗ P (cid:48) Classification = Z deco weak 6 ∗ Z (000) 1168.112 P c Z deco weak { | }∗ Z (000) 1 0169.113 P Classification = Z deco weak 6 Z (006) 1169.115 P (cid:48) Classification = Z deco weak 6 ∗ Z (000) 1169.116 P c Classification = Z deco weak { | }∗ Z (000) 1 0170.117 P Classification = Z deco weak 6 Z (006) 5170.119 P (cid:48) Classification = Z deco weak 6 ∗ Z (000) 1170.120 P c Classification = Z deco weak { | }∗ ∗ Z (000) 1 1171.123 P (cid:48) Classification = Z deco weak 6 ∗ Z (000) 1171.124 P c Classification = Z deco weak { | }∗ Z (000) 1 0172.127 P (cid:48) Classification = Z deco weak 6 ∗ Z (000) 1172.128 P c Classification = Z deco weak { | }∗ ∗ Z (000) 1 1173.131 P (cid:48) Classification = Z deco weak 6 ∗ Z (000) 1173.132 P c Classification = Z deco weak { | }∗ ∗ Z (000) 1 1174.135 P ¯6 (cid:48) Classification = N/Adeco weak ¯6 ∗ P /m (cid:48) Classification = N/Adeco weak 6 i ∗ P /m (cid:48) Classification = N/Adeco weak 6 i ∗ P 622 Classification = N/Adeco weak 6 P (cid:48) (cid:48) Z deco weak 6 ∗ ∗ Z (000) 1 1 0177.152 P (cid:48) (cid:48) Classification = Z deco weak 6 ∗ ∗ Z (000) 1 0 1177.154 P c 622 Classification = Z deco weak { | }∗ Z (000) 1 0 0 0178.155 P 22 Classification = N/Adeco weak 6 P (cid:48) (cid:48) Z deco weak 6 ∗ ∗ Z (000) 1 1 0178.158 P (cid:48) (cid:48) Classification = Z deco weak 6 ∗ ∗ Z (000) 1 0 1178.159 P (cid:48) (cid:48) Classification = Z × Z deco weak 6 ∗ ∗ Z (000) 0 1 1 Z (006) 1 1 0178.160 P c 22 Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 0 1 1179.161 P 22 Classification = N/Adeco weak 6 P (cid:48) (cid:48) Z deco weak 6 ∗ ∗ Z (000) 1 1 0179.164 P (cid:48) (cid:48) Classification = Z deco weak 6 ∗ ∗ Z (000) 1 0 1179.165 P (cid:48) (cid:48) Classification = Z × Z deco weak 6 ∗ ∗ Z (000) 0 1 1 Z (00¯6) ¯5 1 0179.166 P c 22 Classification = Z deco weak { | }∗ ∗ ∗ ∗ Z (000) 1 1 1 1180.167 P 22 Classification = N/Adeco weak 6 P (cid:48) (cid:48) Z deco weak 6 ∗ ∗ Z (000) 1 1 0180.170 P (cid:48) (cid:48) Classification = Z deco weak 6 ∗ ∗ Z (000) 1 0 1180.172 P c 22 Classification = Z deco weak { | }∗ ∗ Z (000) 1 0 0 1181.173 P 22 Classification = N/Adeco weak 6 P (cid:48) (cid:48) Z deco weak 6 ∗ ∗ Z (000) 1 1 0181.176 P (cid:48) (cid:48) Classification = Z deco weak 6 ∗ ∗ Z (000) 1 0 1181.178 P c 22 Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 1 0182.179 P 22 Classification = N/A deco weak 6 P (cid:48) (cid:48) Z deco weak 6 ∗ ∗ Z (000) 1 1 0182.182 P (cid:48) (cid:48) Classification = Z deco weak 6 ∗ ∗ Z (000) 1 0 1182.184 P c 22 Classification = Z deco weak { | }∗ ∗ ∗ Z (000) 1 1 0 1183.185 P mm Classification = Z M deco weak 6 Cm Cm ¯120(0) Z (000) 0 1¯1 1¯1M(100; 0) (000) 0 2¯2 00M(¯120; ) (000) 0 00 2¯2183.187 P (cid:48) m (cid:48) m Classification = Z M deco weak 6 ∗ m ¯120 ∗ Cm Z (000) 1 0 1¯1M(100; 0) (000) 0 0 2¯2183.188 P (cid:48) mm (cid:48) Classification = Z M deco weak 6 ∗ m ∗ Cm ¯120(0) Z (000) 1 0 1¯1M(¯120; ) (000) 0 0 2¯2183.190 P c mm Classification = Z M deco weak { | }∗ Cm Cm ¯120(0) Z (000) 1 0 1¯1 1¯1M(100; 0) (000) 0 0 2¯2 00M(¯120; ) (000) 0 0 00 2¯2184.191 P cc Classification = Z deco weak 6 g ¯12000 g Z (000) 0 1 1184.193 P (cid:48) c (cid:48) c Classification = Z deco weak 6 ∗ g ¯12000 ∗ g Z (000) 1 0 1184.194 P (cid:48) cc (cid:48) Classification = Z deco weak 6 ∗ g ¯12000 g ∗ Z (000) 1 1 0184.196 P c cc Classification = Z deco weak { | }∗ m ¯120 ∗ m ∗ Z (000) 1 0 0 0185.197 P cm Classification = Z M deco weak 6 g ¯12000 Cm Z (000) 0 1 1¯1M(100; 0) (000) 0 0 2¯2185.199 P (cid:48) c (cid:48) m Classification = Z M deco weak 6 ∗ g ¯12000 ∗ Cm Z (000) 1 0 1¯1M(100; 0) (000) 0 0 2¯2185.200 P (cid:48) cm (cid:48) Classification = Z deco weak 6 ∗ g ¯12000 m ∗ Z (000) 1 1 0185.202 P c cm Classification = Z M deco weak { | }∗ ∗ m ¯120 ∗ Cm Z (000) 1 1 0 1¯1M(100; 0) (000) 0 0 0 2¯2186.203 P mc Classification = Z M deco weak 6 g Cm Z (000) 0 1 1¯1M(110; ) (000) 0 0 2¯2186.205 P (cid:48) m (cid:48) c Classification = Z deco weak 6 ∗ m ¯120 ∗ g Z (000) 1 0 1186.206 P (cid:48) mc (cid:48) Classification = Z M deco weak 6 ∗ g ∗ Cm Z (000) 1 0 1¯1M(110; ) (000) 0 0 2¯2186.208 P c mc Classification = Z M deco weak { | }∗ ∗ m ∗ Cm ¯120(0) Z (000) 1 1 0 1¯1M(¯120; ) (000) 0 0 0 2¯2187.211 P ¯6 (cid:48) m (cid:48) ∗ m ¯120 ∗ P ¯6 (cid:48) m (cid:48) Classification = Z M deco weak ¯6 ∗ ∗ Cm Z (000) 0 1 1¯1M(110; ) (000) 0 0 2¯2188.217 P ¯6 (cid:48) c (cid:48) ∗ g ¯12000 ∗ P ¯6 (cid:48) c (cid:48) Classification = Z deco weak ¯6 ∗ g ¯12000 ∗ Z (000) 0 1 1189.223 P ¯6 (cid:48) (cid:48) m Classification = Z M deco weak ¯6 ∗ ∗ Cm Z (000) 0 1 1¯1M(100; 0) (000) 0 0 2¯2189.224 P ¯6 (cid:48) m (cid:48) Classification = N/Adeco weak ¯6 ∗ m ∗ P ¯6 (cid:48) (cid:48) c Classification = Z deco weak ¯6 ∗ ∗ g Z (000) 0 1 1190.230 P ¯6 (cid:48) c (cid:48) Classification = N/Adeco weak ¯6 ∗ g ∗ P /m (cid:48) mm Classification = Z M deco weak 6 ∗ i ∗ ∗ Cm Cm ¯210(0) Z (000) 0 1 0 1 1¯1 1¯1M(100; 0) (000) 0 0 0 0 2¯2 00M(¯210; 0) (000) 0 0 0 0 00 2¯2191.241 P /m (cid:48) m (cid:48) m (cid:48) Classification = N/Adeco weak 6 i ∗ m ∗ P /m (cid:48) cc Classification = Z deco weak 6 ∗ i ∗ ∗ g Z (000) 0 1 0 1 1192.251 P /m (cid:48) c (cid:48) c (cid:48) Classification = N/Adeco weak 6 i ∗ g ∗ P /m (cid:48) cm Classification = Z M deco weak 6 ∗ i ∗ ∗ Cm Z (000) 0 1 0 1 1¯1M(100; 0) (000) 0 0 0 0 2¯2193.261 P /m (cid:48) c (cid:48) m (cid:48) Classification = N/Adeco weak 6 i ∗ m ∗ P /m (cid:48) mc Classification = Z M deco weak 6 ∗ i ∗ ∗ g Cm Z (000) 0 1 0 1 1 1¯1M(110; 0) (000) 0 0 0 0 0 2¯2194.271 P /m (cid:48) m (cid:48) c (cid:48) Classification = N/Adeco weak 6 i ∗ g ∗ P 23 Classification = N/Adeco weak 3 P I 23 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0196.4 F 23 Classification = N/Adeco weak 3 F S 23 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0197.7 I 23 Classification = N/Adeco weak 3 P 12 12 P I Z deco weak { | 12 12 12 }∗ ∗ Z (000) 1 0 1199.12 I P m (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 i ∗ P n (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 12 12 i ∗ F m (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 i ∗ F d (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 14 14 i ∗ Im (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 i ∗ P a (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 12 12 i ∗ Ia (cid:48) ¯3 (cid:48) Classification = N/Adeco weak 3 i ∗ P 432 Classification = N/Adeco weak 3 P (cid:48) (cid:48) Classification = Z deco weak 3 ∗ ∗ Z (000) 0 1 1207.43 P I 432 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0 0 P 32 Classification = N/Adeco weak 3 12 12 12 12 12 12 P (cid:48) (cid:48) Classification = Z deco weak 3 12 12 12 ∗ 12 12 12 ∗ Z (000) 0 1 1208.47 P I 32 Classification = Z deco weak { | 12 12 12 }∗ ∗ ∗ Z (000) 1 0 1 1209.48 F 432 Classification = N/Adeco weak 3 F (cid:48) (cid:48) Classification = Z deco weak 3 ∗ ∗ Z (000) 0 1 1209.51 F S 432 Classification = Z deco weak { | 12 12 12 }∗ Z (000) 1 0 0 0210.52 F 32 Classification = N/Adeco weak 3 14 14 14 14 14 14 F (cid:48) (cid:48) Classification = Z deco weak 3 14 14 14 ∗ 14 14 14 ∗ Z (000) 0 1 1210.55 F S 32 Classification = Z deco weak { | 12 12 12 }∗ 14 14 14 14 14 14 Z (000) 1 0 0 0211.56 I 432 Classification = N/Adeco weak 3 I (cid:48) (cid:48) Classification = Z deco weak 3 ∗ ∗ Z (000) 0 1 1212.59 P 32 Classification = N/Adeco weak 3 34 34 14 14 34 34 P (cid:48) (cid:48) Classification = Z deco weak 3 34 34 14 ∗ 14 34 34 ∗ Z (000) 0 1 1212.62 P I 32 Classification = Z deco weak { | 12 12 12 }∗ 14 14 34 ∗ 34 14 14 ∗ Z (000) 1 0 1 1213.63 P 32 Classification = N/Adeco weak 3 14 14 34 34 14 14 P (cid:48) (cid:48) Classification = Z deco weak 3 14 14 34 ∗ 34 14 14 ∗ Z (000) 0 1 1213.66 P I 32 Classification = Z deco weak { | 12 12 12 }∗ 14 14 34 34 14 14 Z (000) 1 0 0 0214.67 I 32 Classification = N/Adeco weak 3 14 14 ¯ 14 14 14 I (cid:48) (cid:48) Classification = Z deco weak 3 14 14 ¯ ∗ 14 14 14 ∗ Z (000) 0 1 1215.72 P ¯4 (cid:48) m (cid:48) Classification = N/Adeco weak 3 ¯4 ∗ m ∗ F ¯4 (cid:48) m (cid:48) Classification = N/Adeco weak 3 ¯4 ∗ m ∗ I ¯4 (cid:48) m (cid:48) Classification = N/Adeco weak 3 ¯4 ∗ m ∗ P ¯4 (cid:48) n (cid:48) Classification = N/Adeco weak 3 ¯4 12 12 12 ∗ g 12 12 12 ∗ F ¯4 (cid:48) c (cid:48) Classification = N/Adeco weak 3 ¯4 12 12 12 ∗ g 12 12 12 ∗ I ¯4 (cid:48) d (cid:48) Classification = N/Adeco weak 3 ¯4 14 14 ¯ ∗ g 14 14 14 ∗ P m (cid:48) ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ ¯4 ∗ m ∗ P n (cid:48) ¯3 (cid:48) n (cid:48) Classification = N/Adeco weak 3 i ∗ ¯4 ∗ g ∗ P m (cid:48) ¯3 (cid:48) n (cid:48) Classification = N/Adeco weak 3 12 12 12 i ∗ 12 12 12 ¯4 12 12 12 ∗ g 12 12 12 ∗ P n (cid:48) ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ 12 12 ¯4 ∗ g ∗ F m (cid:48) ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ ¯4 ∗ m ∗ F m (cid:48) ¯3 (cid:48) c (cid:48) Classification = N/Adeco weak 3 12 12 12 i ∗ 12 12 12 ¯4 12 12 12 ∗ g 12 12 12 ∗ F d (cid:48) ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ 14 14 ¯4 ∗ g ∗ F d (cid:48) ¯3 (cid:48) c (cid:48) Classification = N/Adeco weak 3 14 12 14 i ∗ 14 14 12 ¯4 14 12 14 ∗ g 14 12 14 ∗ Im (cid:48) ¯3 (cid:48) m (cid:48) Classification = N/Adeco weak 3 i ∗ ¯4 ∗ m ∗ Ia (cid:48) ¯3 (cid:48) d (cid:48) Classification = N/Adeco weak 3 14 14 ¯ i ∗ 14 14 14 ¯4 14 14 ¯ ∗ g 14 14 ¯14