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Jingheng Fu, Mikael Kuisma, Ask Hjorth Larsen, Kohei Shinohara, [Atsushi Togo](https://orcid.org/0000-0001-8393-9766), Kristian S Thygesen

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[Symmetry classification of 2D materials: layer groups versus space groups](https://mdr.nims.go.jp/datasets/c48153ac-101f-45d6-af99-27c9031acee6)

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Symmetry classification of 2D materials: layer groups versus space groups2D Materials      PAPER • OPEN ACCESSSymmetry classification of 2D materials: layergroups versus space groupsTo cite this article: Jingheng Fu et al 2024 2D Mater. 11 035009 View the article online for updates and enhancements.You may also likePeculiar symmetry-protected electronicdispersions in two-dimensional materialsV Damljanovi, N Lazi, A Šolaji et al.-Quantum walk and Anderson localizationof rotational excitations in disorderedensembles of polar moleculesT Xu and R V Krems-Electronic structures near unmovablenodal points and lines in two-dimensionalmaterialsV Damljanovi and N Lazi-This content was downloaded from IP address 144.213.253.16 on 23/04/2024 at 09:09https://doi.org/10.1088/2053-1583/ad3e0c/article/10.1088/1361-648X/abaad1/article/10.1088/1361-648X/abaad1/article/10.1088/1367-2630/17/6/065014/article/10.1088/1367-2630/17/6/065014/article/10.1088/1367-2630/17/6/065014/article/10.1088/1751-8121/accf51/article/10.1088/1751-8121/accf51/article/10.1088/1751-8121/accf512D Mater. 11 (2024) 035009 https://doi.org/10.1088/2053-1583/ad3e0cOPEN ACCESSRECEIVED25 December 2023REVISED5 April 2024ACCEPTED FOR PUBLICATION12 April 2024PUBLISHED22 April 2024Original Content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PAPERSymmetry classification of 2D materials: layer groups versusspace groupsJingheng Fu1, Mikael Kuisma2, Ask Hjorth Larsen2, Kohei Shinohara3, Atsushi Togo4and Kristian S Thygesen2,∗1 State Key Laboratory of Low Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, People’sRepublic of China2 CAMD, Computational Atomic-Scale Materials Design, Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby,Denmark3 Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan4 Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan∗ Author to whom any correspondence should be addressed.E-mail: thygesen@fysik.dtu.dkKeywords: layer groups, 2D materials, database, high-throughputAbstractThe symmetry of a crystal structure with a three-dimensional (3D) lattice can be classified by oneof the 230 space group types. For some types of crystals, e.g. crystalline films, surfaces, or planarinterfaces, it is more appropriate to assume a two-dimensional (2D) lattice. With this assumptionthe structure can be classified by one of the 80 layer group types. We have implemented analgorithm to determine the layer group type of a 3D structure with a 2D lattice, and applied it tomore than 15 000 monolayer structures in the Computational 2D Materials Database (C2DB). Wecompare the classification of monolayers by layer groups and space groups, respectively. The latteris defined as the space group of the 3D bulk structure obtained by repeating the monolayerperiodically in the direction perpendicular to the 2D lattice (AA-stacking). By this correspondence,nine pairs of layer group types are mapped to the same space group type due to the inability of thespace group to distinguish the in-plane and out-of-plane axes. In total 18% of the monolayers inthe C2DB belong to one of these layer group pairs and are thus not properly classified by the spacegroup type. Our results show that symmetry classification of 2D materials should be based on layergroups rather than the commonly used space groups.1. IntroductionThe classification of periodic solids by means of sym-metry groups is fundamental to our understandingand description of crystalline materials. In additionto providing a unique classification scheme for crystalstructures, the symmetry group can be used to deriveimportant qualitative information about the physicalproperties of a material before they are measured orcalculated. For example, the symmetry group dictatesthe possible degrees of degeneracy of energy levels,governs the selection rules for electronic and vibronictransitions [1], determines the possible topologicalphases [2], and relates different spatial componentsof response tensors of crystals [3].The symmetry of a three-dimensional (3D) crys-tal structure with 3D translation periodicity, i.e. a 3Dlattice, is described by one of the 230 space grouptypes (from hereon we shall for simplicity use themore common term ‘space group’ in place of ‘spacegroup type’, see appendix A.4). For a 3D crystal struc-ture with a two-dimensional (2D) lattice, for examplea crystalline film, the relevant symmetry group is rep-resented by the layer group. There exist a total of 80layer groups in contrast to the 230 space groups. Forthe point group operations of the layer group, thesymmetry axis of the operations must be either paral-lel or perpendicular to the 2D lattice plane.Moreover,the direction of the translation part of the symmetryoperations must be parallel to the lattice plane.Materials with a 2D lattice and a thickness of onlya few atoms are generally referred to as 2D mater-ials. This class of materials has attracted enormousattention from a broad range of scientific communit-ies due to their unique and easily tunable proper-ties, their extreme thinness, and their good device© 2024 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/2053-1583/ad3e0chttps://crossmark.crossref.org/dialog/?doi=10.1088/2053-1583/ad3e0c&domain=pdf&date_stamp=2024-4-22https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0009-0002-6653-4155https://orcid.org/0000-0002-5907-2549https://orcid.org/0000-0001-5197-214Xmailto:thygesen@fysik.dtu.dk2D Mater. 11 (2024) 035009 J Fu et alintegration potential [4–8]. Despite of this interest,the symmetry characterization of 2D materials hasnot received the attention it deserves, and is oftenrestricted to specification of the point group or thespace group of some related layered bulk structures.This holds in particular for all the computationaldatabases of 2Dmaterials, e.g. the Computational 2DMaterials Database (C2DB) [9, 10], the MaterialsCloud [11], and the 2DMatPedia [12], which holdthe atomic structures of thousands of monolayers.Similar to the vastmajority of studies on 2Dmaterials,these databases have classified the monolayers by thespace group of the bulk structure obtained by repeat-ing the monolayer periodically in the direction nor-mal to the 2D lattice plane, i.e. the space group of theAA-stacked bulk crystal. An important reason for thisstate of affairs has been the lack of a readily accessiblesoftware tool to determine the layer group of a general2D atomic structure.Here we report on a numerical algorithm todetermine the layer group of a 3D crystal structurewith a 2D lattice. The algorithm has been implemen-ted and made available as part of the open sourcespace group and magnetic space group symmetryanalysis library Spglib [13, 14], which is widely usedin the computational materials science community.We use the algorithm to obtain the layer groups ofthe 15 733 crystal structures currently contained inthe C2DB database. A statistical analysis of the datareveals a rather inhomogeneous distribution with anumber of layer group types being either very scarcelyrepresented or not represented at all. We also showthat the classification of a monolayer by the spacegroup of its AA-stacked bulk structure is incomplete.In fact, for 18% of the monolayers in C2DB, thespace group of the AA-stacked bulk crystal does notuniquely determine the layer group.2. MethodIn this paper, we use layer system to refer to the collec-tion of a layer group crystallographic coordinate sys-tem and the atomic structure defined in this coordin-ate system. Following [13, 15], a, b and c representbasis vectors of the system, with two of the basis vec-tors being lattice vectors. Without loss of generality,we take the lattice vectors to be a and b. Furthermore,we denote the basis vectors in different unit cells by(ai,bi,ci), (ap,bp,cp), and (ac,bc,cc), where the sub-scripts refer to the input cell, primitive cell, and con-ventional cell, respectively. Atom coordinates in theunit cell basis, i.e. fractional coordinates, are denotedby x. When necessary, the subscript i, p or c is usedto indicate the type of unit cell. For layer groups,the role of c is only to provide a basis vector forrepresenting the atomic positions. Nevertheless, thecurrent implementation follows the crystallographicspace group convention which requires the rotationpart of symmetry transformations to be representedby integer matrices under the primitive cell basis (seesection 2.2). Hence the choice of c-basis can affect thepoint group and thus layer group. In this paper, we fixthe non-lattice vector ci to be orthogonal to ai and bito obtain the highest symmetry.For crystals with periodicity along the three basisaxes, the distance between two atoms with positionsx and x ′ respectively is defined byd(x,x ′) = ∥(a,b,c)(∆x−⌊∆x⌉)∥. (1)Here (a,b,c) denotes the coordinate matrix of theunit cell basis vectors, ∆x= x− x ′, and ⌊·⌉ roundscomponents of a vector to the nearest integer. Layersystems do not have periodicity in the direction alongthe non-lattice vector c, and the rounding opera-tion in equation (1) is not applied to this coordinate.To account for small distortions or numerical noise,Spglib determines the equality of two atom posi-tions by the conditiond(x,x ′)< ϵ, (2)where ϵ is a small tolerance parameter (called ‘sym-prec’) set by the user.A symmetry operation maps the crystal structureonto itself. The operation is denoted (W,w), wherethe linear part W is a 3× 3 integer matrix, and thetranslation part w is a 3× 1 column vector. By thesymmetry operation, an atom with position x will besent tox̃=Wx+w. (3)Symmetry requires the position x ′ of one and onlyone atomwith the same species to be equal to x̃ in thesense of equation (2).Our algorithm for determining the layer group ofa layer system is illustrated in figure 1. The algorithmfollows the one already implemented in Spglib forspace groups and documented elsewhere [13]; thuswe mainly focus on the aspects that are specific tolayer groups. The lower part of figure 1 shows themain steps required to determine the layer/spacegroup while the upper part details the criteria used toselect the conventional unit cell for the various crystalsystems. In addition to determining the point groupand space/layer group, the algorithm also determinesthe site symmetry group for each atom and the occu-pied Wyckoff positions, and symmetrizes the inputstructure.2.1. Primitive cellThe input cell will be transformed to a provisionalprimitive cell to reduce the computational complex-ity. To determine the primitive cell, the first step isto check whether the input cell can be reduced to asmaller unit that generates the crystal. Specifically, thealgorithm searches for symmetry operations that arepure translations, i.e. Wi = I and wi ̸= 0. The three22D Mater. 11 (2024) 035009 J Fu et alFigure 1. Upper: the criteria (rotation axes and metric conditions) used by Spglib to determine a standardized conventional cellof a bulk crystal (3D lattice) and a layer system (2D lattice). The criteria are listed for each of the seven crystal systems. The layersystem can be additionally classified by the Bravais system of its lattice, which is listed in the last row of the table. For triclinic andmonoclinic crystal systems, the criteria specific to bulk and layer systems are listed in the second and third rows, respectively.Monoclinic crystal systems with oblique and rectangular lattices have different criteria, and are separately listed with differentcolors. For orthorhombic systems, the bulk and layer systems share most of the conditions. For the tetragonal, trigonal andhexagonal crystal systems, the bulk and layer systems share all the same criteria, and the rows for these systems are combined. Thecubic crystal system does not exist for layer groups, and is left blank in the table. The criteria include axis choices for selecting ac,bc and cc from the symmetry operation with certain types, and metric parameters. The remaining conditions are additionalmetric rules used by Spglib when the standard choices defined in [15, 18] cannot uniquely determine the shape and/or setting ofthe conventional cells. Details of the rules are described in appendix C. Lower: workflow used by the algorithm to determine thepoint group, space/layer group, Wyckoff positions, and more.vectors among the foundws and the input cell basis ai,bi and ci which form the cell with smallest non-zerovolume are the primitive cell basis vectors. For layersystems, the non-lattice vector ci will always be chosenas the primitive basis vector cp. The obtained primit-ive cell will further be transformed to the Delaunaycell [16]. The shape of the Delaunay primitive cellbelongs to one of the 24 symmetric varieties [16, 17].Matrix elements of the linear partWp are 0, 1 or −1under the Delaunay reduced basis. Note that this pro-cedure may not be fully applicable to a layer system asit can transform the lattice vectors into linear combin-ations of the original lattice and non-lattice vectors.Thus the Delaunay reduction requires special care forlayer systems as discussed in detail in appendix B.2.2. Symmetry operationsAfter obtaining the primitive cell, the algorithmsearches for the matrix-column pairs (Wp,wp) of allsymmetry operations in the Delaunay reduced prim-itive cell basis (ap,bp,cp). For space group operations,Wp consists of 0 and±1 satisfying det(Wp) =±1. Forlayer groups, preservation of the 2D lattice prohibitsmixing the non-periodic component into the latticevectors, e.g. a ′ = a+ c. For layer systems, ‘A’-face, ‘B’-face, body-, and face centered conventional cells donot exist. Consequently, these conventional cells arenot considered. the algorithm searches for operationswith the matrix patternWlayer =W11 W12 0W21 W22 00 0 ±1 Wij = 0,±1. (4)With this matrix pattern, any operation with rotationorder higher than twomust have the rotation axis per-pendicular to the lattice plane. Consequently, pointgroups belonging to the cubic crystal system are notallowed for layer systems. In addition, any ±2-foldrotations must have the rotation axis either parallelor perpendicular to the lattice plane. Furthermore,equation (1) rules out the possibility of finding screwrotation or glide reflection with a non-lattice com-ponent in the translation part w. To find the sym-metry operations of a layer system, the algorithmsearches for all transformations (W,w) with W ofthe form (4) satisfying d(x̃,x ′)< ϵ, where d(·, ·) isdefined in equation (1), x̃=Wx+w, and x and x ′are the position of atoms of the same type.2.3. Point groupThe linear partW of all found symmetry operations(W,w) compose the crystallographic point group Pin the primitive cell basis. The type of the pointgroup is determined by counting the number of the32D Mater. 11 (2024) 035009 J Fu et alten different rotation orders (±1,±2,±3,±4,±6)appearing in P (see table V in [13]). Once the pointgroup type has been determined, the search for thespace/layer group can be narrowed down accordingly.2.4. Standardized conventional cellTo determine the space/layer group type from thesymmetry operations they must be compared withthose already stored in Spglib. However, a directcomparison requires that the same crystallographiccoordinate systems (CCSs), i.e. origin position andbasis vectors (including their order), are employed.For each space group, reference [18] lists the possibleconventional CCSs and designates one of them as thestandard choice. A standard CCS for layer systems isdefined by [15].Spglib adopts Hall symbols [19] to distin-guish the different choices of CCS. A table of all theHall symbols for space groups can be found in tableA1.4.2.7 of [20]. Spglib contains a database withall (W,w) matrix-column pairs generated using Hallsymbols. For layer groups, we are not aware of a com-plete table of Hall symbols. Thus we have producedone corresponding to the different choices of CCSslisted in [15], see appendix D.2.5. Layer groupThe crystallographic point group determines the typeof the Bravais lattice and puts some constraints on themetric parameters of the conventional cell (relationsbetween basis vector lengths and angles), see upperpanel of figure 1 and appendixC.Generally, one of thebasis vectors ac, bc and cc is chosen as the symmetryaxes of a specific symmetry operation, as shown inthe first line for each criteria in figure 1. The shortesttwo vectors in the perpendicular plane that satisfy themetric conditions are chosen for the remaining twobasis vectors. TheW matrices will retain integer ele-ments under this basis.When the basis vector choices are the same asthe standard CCS of the relevant space/layer group,W will match the standard counterparts stored inSpglib. Still, w may differ from the stored ones by ashift of the origin. The origin shift is determined fol-lowing the algorithm in [13, 21] except the distancesin the algorithm are measured by equation (1).2.6. SymmetrizationThe site symmetry group of each atom is determinedusing the distance measure in equations (1) and (2).Each atom is further assigned a Wyckoff position.Finally, the atomic structure is symmetrized by apply-ing all of the site symmetry operations separately toeach atom and performing an average over the res-ulting positions. The standardized conventional cell issymmetrized such as to satisfy the lattice metric con-ditions in figure 1(upper) exactly to obtain the sym-metrized input cell (as,bs,cs) and the standard prim-itive cell. For the standard conventional cells withprimitive lattice, the standard primitive cells are theconventional cells. While for ‘C’-face centered stand-ard conventional cells, the standard primitive cell(asp,bsp,csp) = (asc,bsc,csc)PC, wherePC = 1212 01̄212 00 0 1 . (5)3. ResultsThe C2DB currently contains 15 735 2D materi-als. Most of the structures have been generated bycomputational exfoliation of experimentally knownlayered bulk crystals followed by systematic latticedecoration (atomic substitution) of the thus obtainedmonolayers [9, 10]. Recently, the data set wasamended by monolayers created by a deep generat-ive model [22]. All the monolayers in C2DB havebeen relaxed using density functional theory with thePerdew Burke Ernzerhof (PBE) exchange-correlationfunctional [23].We have determined the layer group of all themonolayers in C2DB alongside the space group ofthe corresponding AA-stacked bulk structures (seenext section). Figure 2 shows the distribution of theC2DB monolayers according to the layer group. Thedistribution shown has been limited to the subset ofmonolayers for which the band gap has been calcu-lated (7201 materials), and the materials have beendivided into metallic and non-metallic compounds.The background color is used to indicate the differentcrystal systems (along with Bravais system if neces-sary). Examples of crystal structures from selectedlayer groups are shown.The most frequently occurring layer group isnumber 72 (p3̄m1) with 1231 occurrences. It shouldbe noted, that the distribution of materials by layergroup number is highly non-uniform, with severalcompletely empty bins accompanied by several withjust a few materials. In particular, there are no mater-ials in C2DB with layer group numbers 19, 24, 25, 30,38, 39, 43, 49, 54, 56, 60, 73, 75, 76 and 77 (using atolerance parameter ‘symprec’= 0.1 Å).There is a tendency that insulating behavior (bluebars) is more pronounced among the materials withlower layer group numbers while metallic behavior(red bars) is dominant in materials with high layergroup numbers. This is due to the fact that, as a roughrule, layer groupswith smaller numbers contain fewersymmetry transformations. The less symmetric crys-tals will have energy bands of lower degeneracy and42D Mater. 11 (2024) 035009 J Fu et alFigure 2. Distribution of materials of C2DB according to layer group number. The background color coding refers to the crystalsystem (the monoclinic crystal system is further classified according to its lattice type). Some examples of structures fromdifferent layer groups are shown. The blue (red) part of the bars refer to materials with finite (zero) band gap.therefore a higher chance of all bands being eitherfully occupied or empty. On the other hand, moresymmetric crystals will have energy bands of higherdegeneracy and consequently a higher chance of par-tially occupied, i.e. metallic, bands.3.1. Layer group versus space groupWhen discussing the layer group of a 2D mater-ial, we model the material as an ideal crystal withfinite thickness along the c direction and infinite peri-odicity along the a and b directions, as shown infigure 3(a). On the other hand, as mentioned in theintroduction, 2D materials have so far mainly beenclassified using space groups. The use of space groupsimplies a 3D lattice and thus an artificial periodicitymust be imposed along the c direction. Therefore,the space group actually represents the symmetry ofa 3D crystal consisting of the 2D material repeatedperiodically along the c direction. For simplicity, weassume c to be perpendicular to the ab lattice planeand refer to the resulting bulk crystal as the ‘AA-stacking model’, see figure 3(b). When a sufficientlylarge vacuum region is used to separate the 2D lay-ers along c, the AA-stackingmodel corresponds to thestructures used to simulate 2D materials in periodicplane wave density functional theory calculations.Perhaps for this reason the space group of the AA-stacking model has been widely used to represent thesymmetry of 2Dmaterials. In this sectionwe comparethe two classification schemes and highlight possiblepitfalls arising when using the AA-stacking modeland the corresponding space group to describe a 2Dmaterial.Figure 3. (a) The 2D material is periodic along the a and baxes, but not along the c axis. (b) In the AA-stacking model,the 2D material is periodically repeated along theout-of-plane c axis, extending it to the domain of spacegroups and making it compatible with computational planewave codes, which require periodic boundary conditions inall directions.52D Mater. 11 (2024) 035009 J Fu et alFigure 4. Examples of structures that are incompletelyclassified by the space group of their AA-stacked bulk,see table 1. Each row shows a pair of 2D crystalstructures belonging to distinct layer groups, but withthe same space group (of the AA-stacked bulk).Structures from the layer group pair (24, 31) are notshown as the C2DB contains no monolayers from layergroup 24.Table 1. List of all cases where two different layer groups aremapped to the same space group upon AA stacking of the 2Dmaterial. The number of monolayers in the C2DB is indicated inparenthesis after the Hermann–Mauguin symbol of each layergroup with tolerance parameter set to 10−1 Å. In figure 4 someillustrative examples of crystal structures from the relevant layergroups are shown.Layer group pairs Space group3: p112 (46) 8: p211 (11) 3: P24: p11m (59) 11: pm11 (591) 6: Pm5: p11a (201) 12: pb11 (11) 7: Pc6: p112/m (30) 14: p2/m11 (230) 10: P2/m7: p112/a (51) 16: p2/b11 (88) 13: P2/c23: pmm2 (375) 27: pm2m (186) 25: Pmm228: pm21b (245) 29: pb21m (24) 26: Pmc2124: pma2 (0) 31: pm2a (10) 28: Pma240: pmam (5) 41: pmma (154) 51: PmmaFirst, on a general note, it has been shown that foreach layer group Lwith point groupP and translationlattice LL, there exists a symmorphic representativewhich is the 3D space group S that shares the samepoint group and symmetry diagram [24, 25]. The lat-tice of the symmorphic representativeLS = LL ⊕ LR (6)is the direct sum of the 2D lattice of the layer groupL and the 1D lattice LR of a symmorphic rod groupR with point group P . The layer group and its sym-morphic representative can be represented by thesame Hermann–Mauguin symbol which only differby the upper and lowercase of the first letter [25].However, for some types of the space groups, the split-ting of the lattice LS into two P-invariant sub-latticesis not unique. Consequently, one symmorphic rep-resentatives may correspond to multiple layer groupswith different types. Ambiguity thereupon arises.To be more specific, the symmorphic space grouprepresentatives of the 80 layer groups belong to 71 dif-ferent space groups. Only 62 of these 71 space groupshave a one-to-one mapping with the correspondinglayer groups, while each of the remaining nine spacegroups are related to two different layer groups. Intable 1, we list these nine non-injective mappings.The numbers in parenthesis represent the numberof materials in C2DB with the specific layer group.Examples of concrete structures from the relevantlayer groups are provided in figure 4. Note that onlyeight cases are shown in figure 4 as C2DB currentlydoes not include any structures from layer group 24.As illustrated in table 1, a total of 2848 of the mono-layers in C2DB (18%) belong to one of the layergroups in table 1, and thus are not uniquely classifiedby the space group of the AA-stacked structure.Inspection of the structures/layer groups infigure 4 and table 1 shows that the first five pairs62D Mater. 11 (2024) 035009 J Fu et alare examples where the two layer groups belong tothe monoclinic crystal systems with oblique and rect-angular Bravais systems, respectively. In these casesone of the structures has an in-plane ±2-fold sym-metry, which is replaced by an out-of-plane ±2-foldsymmetry in the other structure. The last four pairsof layer groups all belong to the orthorhombic crystalsystem. For each pair of the examples, the out-of-plane rotation,mirror reflection or glide translation isexchanged with an in-plane symmetry that has a dif-ferent type. For example, the out-of-plane two-foldrotation and one of the in-plane mirror reflectionsof layer group 23 (pmm2) becomes an out-of-planemirror reflection and in-plane two-fold rotation forlayer group 27 (pm2m). These distinctions cannot bemade by the space group of the AA-stacking model.The ability of the layer groups to differentiatestructures of the same space group (defined via theAA-stacking model) is not without practical value.For example, Ji et al showed how layer groups canbe used as a tool to determine possible outcomes offerroelectricity when two monolayers are stacked toform a bilayer [26]. They further divided the layergroups into distinct polar types, according to whetherpolarization is allowed in-plane, out-of-plane, orboth. An example is given by figure 4, where layergroup 4 (p11m) is polar in-plane and layer group 11(pm11) is polar out-of-plane, while both share thespace group 6 (Pm). Many other properties of 2Dmaterials, e.g. classification of electronic [27, 28]and topological [29–31] phases are also governed bythe layer group symmetry. We note that useful crys-tallographic and Brillouin-zone databases for layergroups are available on the Bilbao crystallographicserver [32].The results of this section show that a symmetryclassification of 2D materials based on space groups,such as the AA-stacking model, is less precise andmore coarse grained than one based on layer groups.Consequently, to ensure a more accurate and unam-biguous description of 2D crystal symmetries it is ourrecommendation that the 2D materials communityabandons the use of space groups and uses layergroups instead.4. ConclusionsWe have introduced an algorithm to determine thelayer group of a crystal structure with a 2D periodiclattice, and applied it to 15 000+ atomically thin crys-tals stored in the C2DB. A symmetry analysis of the2D materials in C2DB revealed a rather inhomogen-eous distribution of layer groups with several layergroups only sparsely populated and 15 groups notrepresented at all. It would be interesting for future2D materials discovery projects to explore crystalsfrom these layer groups.We compared two different symmetry classific-ation schemes for 2D materials based, respectively,on the layer group of the 2D crystal and the spacegroup of the 3D bulk crystal obtained by repeatingthe 2D material periodically in the direction normalto the 2D plane (AA-stacking model). We showedthat because the latter scheme does not distinguishbetween the in-plane and out-of-plane directions, it isless accurate and unable to distinguish between some2D crystals with different layer groups. This problemoccurs for 18% of the crystals in the C2DB. On thisbasis we conclude that symmetry classification of 2Dmaterials should be based on layer groups rather thanspace groups.The method to determine the layer group is avail-able as part of the open source Spglib library.The algorithm will be useful for researchers workingwith 2D crystal structures in general, and will facilit-ate the application of various theories and conceptsdeveloped for 2D materials that build on the layergroup symmetry.Data availability statementThe data that support the findings of this study areopenly available at the following URL/DOI: https://github.com/spglib/spglib.AcknowledgmentWe acknowledge funding from the EuropeanResearch Council under the European Union’sHorizon 2020 Research and Innovation ProgramGrant No. 773122 (LIMA) and Grant AgreementNo. 951786 (NOMAD CoE). This work was alsosupported by the Basic Science Center Project ofNSFC (Grant No. 52388201). K S T is a VillumInvestigator supported by VILLUM FONDEN(Grant No. 37789).Appendix A. Classification of layer groupBelow we summarize the mathematical structure andclassification of the layer groups. We mainly followthe terminologies in references [15, 18]. We clas-sify the layer groups according to the point groups(appendix A.1), Bravais lattice (appendix A.2), andboth (appendix A.3).A.1. Point group of layer groupTwo point groups,P andP ′, belong to the same geo-metric crystal class if they are conjugate by an invert-ible matrix P, i.e. P−1PP= P6 ′. Two layer groups,7https://github.com/spglib/spglibhttps://github.com/spglib/spglib2D Mater. 11 (2024) 035009 J Fu et alTable 2. Classification of the point groups of layer groups. Namesand symbols of the crystal systems and geometric crystal classesare based on table 3.2.1.4 of [18]. The Laue class classifies thegeometric crystal classes by ignoring inversion.Crystal system Laue class Geometric crystal classesTriclinic 1̄ 1, 1̄Monoclinic 2/m 2,m, 2/mOrthorhombic mmm 222, 2mm,mmmTetragonal 4/m 4, 4̄, 4/m4/mmm 422, 4̄2m, 4mm, 4/mmmTrigonal 3̄ 3, 3̄3̄m 32, 3m, 3̄mHexagonal 6/m 6, 6̄, 6/m6/mmm 622, 6̄2m, 6mm, 6/mmmL and L ′, with point groups P and P ′ belong to thesame geometric crystal class if and only if P and P ′belong to the same geometric crystal class. Becausethe point groups of layer groups are 3D, geometriccrystal classes of layer groups are classified with thesame notation as those of space groups.Two point groups belong to the same crystal sys-tem if and only if the sets of Bravais type of latticeswhich these point groups leave invariant, coincide.Crystal systems of layer groups are also classified withthe same notation as those of space groups: triclinic,monoclinic, orthorhombic, tetragonal, trigonal, andhexagonal. Note that a cubic crystal system does notexist for layer groups because the point groups of layergroups should preserve the 2D lattice plane (the pointgroup of the cubic crystal system would contain anoperation that rotates the structure out of the lat-tice plane). Table 2 shows the classification of pointgroups for layer groups.A.2. Translation lattice of layer groupFor a lattice L, a Bravais group B(L) is a set of iso-metrymappings that preserve L. Two lattices L and L ′belong to the same Bravais type of lattices if and onlyif their Bravais groups are conjugate by a unimodularmatrixP, i.e.P−1B(L)P= B(L ′). Because translationlattices of layer groups are two-dimensional, Bravaistypes of lattices of layer groups are classified with thesame notation as those of plane groups.Two lattices L and L ′ belong to the same latticesystem if and only if their Bravais groups belong tothe same geometric crystal class. The correspondingTable 3. Classification of translation lattices of layer groups.Names and symbols of lattice systems are based on table 3.1.1.1 of[18]. Symbols of Bravais types of lattices are based on table 3.1.2.1of [18].Lattice system Holohedry Bravais typeOblique (monoclinic)m 2/m mpRectangular mmm op(orthorhombic) o ocSquare (tetragonal) t 4/mmm tpHexagonal h 6/mmm hppoint group for the geometric crystal class is calleda holohedry. Lattice systems of layer groups are alsoclassified with the same notation as those of planegroups: oblique, rectangular, square, and hexagonal.Table 3 shows the classification of translation latticesof layer groups.A.3. Arithmetic-geometric crystal classLet Ti and Pi be a translation subgroup and a pointgroup of layer groups Li (i = 1,2). Two layer groupsL1 and L2 belong to the same arithmetic-geometriccrystal class [33] ifP1 andP2 are conjugate by a basistransformation from T1 to T2 and a transformationalong the non-periodic axis. Table 4 shows the classi-fication of layer groups by the arithmetic-geometriccrystal classes.A.4. Layer group or layer group type?Layer group type defines the equivalence relationshipbetween different layer groups. Two layer groups Land L ′ belong to the same layer group type if they areconjugate by an orientation-preserving transforma-tion (P,p), i.e. (P,p)−1L(P,p) = L ′ and det(P)> 0.The terminologies ‘group’ and ‘group type’ areoften mixed by material scientists. For example, sen-tence like ‘the layer group of graphene is p6/mmm’is commonly used in research paper. However, thisis not accurate, because p6/mmm is in fact the typeof the layer group. The layer group is the group con-taining six-fold rotation, mirror reflection, transla-tion symmetry and etc. The layer group of a graphenebefore and after we stretch it isotropically are dif-ferent, because the lattice constant changes, and thetranslation symmetry in the group alters accordingly,while its layer group type keeps to be p6/mmm.82D Mater. 11 (2024) 035009 J Fu et alTable 4. Arithmetic-geometric crystal classes for layer groups. Each arithmetic-geometric crystal class is represented by a layer group inITE whose sequential number is the smallest among the belonging arithmetic-geometric crystal class.Crystal class/lattice system Geometric crystal class Arithmetic-geometric crystal classTriclinic/oblique 1 p1 (1)1̄ p1̄ (2)Monoclinic/oblique 2 p112 (3)m p11m (4)2/m p112/m (6)Monoclinic/rectangular 2 p211 (8)c211 (10)m pm11 (11)cm11 (13)2/m p2/m11 (14)c2/m11 (18)Orthorhombic/rectangular 222 p222 (19)c222 (22)mm2 pmm2 (23)cmm2 (26)(m2m) pm2m (27)cm2m (35)mmm pmmm (37)cmmm (47)Tetragonal/square 4 p4 (49)4̄ p4̄ (50)4/m p4/m (51)422 p422 (53)4mm p4mm (55)4̄2m p4̄2m (57)(4̄m2) p4̄m2 (59)4/mmm p4/mmm (61)Trigonal/hexagonal 3 p3 (65)3̄ p3̄ (66)312 p312 (67)(321) p321 (68)3m1 p3m1 (69)(31m) p31m (70)3̄1m p3̄1m (71)(3̄m1) p3̄m1 (72)Hexagonal/hexagonal 6 p6 (73)6̄ p6̄ (74)6/m p6/m (75)622 p622 (76)6mm p6mm (77)6̄m2 p6̄m2 (78)(6̄2m) p6̄2m (79)6/mmm p6/mmm (80)Appendix B. Delaunay reductionGiven an n-dimensional (nD) lattice, the goal ofDelaunay reduction is to find the shortest basis vec-tors bi(1⩽ i ⩽ n) spanning the lattice. In practice thealgorithm achieves that by minimizingn+1∑i=1b2i , (B1)wherebn+1 =−n∑i=1bi. (B2)Forn= 2, we start by selecting twobasis vectorsb1and b2. The extended basis b3 =−b1 − b2. Check thethree scalar products bi · bj(i ̸= j) one after the other.Any time bi · bj > 0, the transformation92D Mater. 11 (2024) 035009 J Fu et alFigure 5. Schematic diagram of 2D Delaunay reduction. The solid arrows denote the vectors at an intermediate step during theDelaunay reduction. Since the angle between b2 and b3 is smaller than 90◦, the next step will transform b1 and b2 to b1 + 2b2 and−b2, respectively. The transformed vectors are denoted by dotted arrows. If the 2D lattice is spanned by b1 and a vectorperpendicular to the paper, such transformation breaks the lattice plane and is not allowed. The Delaunay reduction will stop atthe previous step. The above mentioned case appears when b2 lies in the white circles. The Delaunay reduction can be fully appliedwhen b2 lies in the rest areas (edge included). Green and yellow indicate b1 finally being transformed to b1 and−b1, respectively.b ′i =−bib ′j = bjb ′k = bk + 2bi(B3)is performed, where k ̸= i, j. After the transform-ation,∑b ′i ′2=∑b2i ′ − 4bi · bj <∑b2i ′ . Then wecheck b ′1 · b ′2, b′1 · b ′3 and b′2 · b ′3, and so on. The loopstops when all bi · bj ⩽ 0, i.e. the angles betweenb1,b2,b3 all become non-acute. 2D Delaunay reduc-tion is applied to the basis vectors forming the obliqueface of a monoclinic cell and works well on bulk sys-tems. For the monoclinic/oblique cell, the target vec-tors to be reduced are the lattice vectors a and b,and the procedure has no difference from the bulkcase. However, the oblique face of a monoclinic/rect-angular cell is composed of b1 = b and b2 = c. Asshown in figure 5, the basis vectors of the periodicplane are b1 and the unique axis a which is vertical tothe paper. Allowed transformations for b1 are b1 →b1 and b1 →−b1. When the angle between b2 andb3 is acute, Delaunay reduction requires b1 → b1 +2b2 and b2 →−b2. The new vectors break the lat-tice plane, which is not permitted (dotted arrows infigure 5). For certain input cell, candidates for b2,b3are predetermined (represented by hollow points infigure 5). When the points lay on the colored areas(including edges), ∀bi · bj ⩽ 0 fulfills after Delaunayreduction. b2 with initial position at the green areaswill be sent to the green area of region C. b2 in theyellow areas will finally deposit in region B, with b1being switched to −b1. If the input b2 lies in any ofthewhite circles, at least one angle between the vectorswill be acute. In Spglib, it is the one between b2 andb3. After reduction, the non-lattice vector c and theshortest two vectors among b1, b2 and b3 are selectedto build the bulk or monoclinic/oblique cell. Whilefor a monoclinic/rectangular cell, one of±b1 must bechosen to maintain the periodic plane. The other twovectors are a and the shortest in b2 and b3.The workflow for n= 3 is similar. The transform-ation for bi · bj > 0 (i, j = 1, . . . ,4; i ̸= j) readsb ′i =−bib ′j = bjb ′k = bk + bib ′l = bl + bi.(B4)For layer systems, when the non-lattice vector clies in the intersection of an infinitely long prismand a sphere, the Delaunay reduction cannot be fullyapplied. The angle between b3 and b4 will be less than90◦. The two basis vectors that form the cross sectionof the prism arep1 =(b1 + b2) · b2(b2 × b1) · ez(ez × b1) and,p2 =(b1 + b2) · b1(b2 × b1) · ez(ez × b2) ,(B5)where ez = (0,0,1), p1⊥b1, p2⊥b2, and p1 + p2 =−b1 − b2. The prism is infinite along the ±z direc-tion. The origin of the sphere is− 12 (b1 + b2) and theradius is− 12 |b1 + b2|.Appendix C. Conventional cellsThe standard crystallographic coordinate system(CCS) choices defined in [15, 18] impose constraintson the choices and the orientations of the basis vec-tors, and are firstly consulted for building the conven-tional cells. Additional metric rules are added whenthe standard choices cannot uniquely determine thecell. The criteria used by Spglib are summarized infigure 1.102D Mater. 11 (2024) 035009 J Fu et alFor triclinic lattices, a Niggli cell [34] is chosento be the conventional cell by virtue of its unique-ness. For bulk crystals (3D lattice), the Niggli condi-tions include |ac|⩽ |bc|⩽ |cc| and |ac · cc|⩽ |ac · bc|if |bc|= |cc| for bulk systems. The Niggli reductionwill swap bc and cc when the aforementioned con-ditions are not met. For layer systems with the non-lattice vector cc, such action is not allowed. In thiscase, the resulting cell will thus differ from the con-ventional Niggli cell by a rotation.The monoclinic crystal system is subdividedinto monoclinic/oblique and monoclinic/rectangularaccording to the 2D Bravais lattice. The unique two-fold axis is vertical to the plane and set to be parallelto cc for the former, while for the latter the axis is in-plane and chosen to be along ac. 2D Delaunay reduc-tion is conducted for the oblique face of the cell. Asshown in appendix B, this procedure cannot be com-pletely performed for themonoclinic/rectangular sys-tem. In this case, both the shape and the orientationof the resulting cell will differ from the conventionalcell that would have been obtained for a monoclinicbulk crystal.The conventional cells of the orthorhombicgroups are cuboids. The standardCCS choice requiresthe axes of certain symmetry operations to be per-pendicular to the faces of the cuboids. Whenmultiplefaces share same symmetry diagram up to an originshift, permutation of the cell vectors will not influencethe CCS choice. To find out which basis vectors can beswapped, the Euclidean normalizer of the space/layergroup is utilized. The Euclidean normalizer of a groupG is the group NE consisting of all Euclidean trans-formations which map G onto itself by conjugationNE (G)≡{s ∈ E | s−1Gs= G}, (C1)where E is the group of all Euclidean transforma-tions [18]. G is a normal subgroup of NE, and thefactor groupNE(G)/G classifies the transformations.Each element sG of the factor group represents a dif-ferent kind of CCS transformation (axis permutation,origin shift, etc) that does not alter the CCS choice.We are interested in all the sG which contains one ofthe six right-handed axis permutations abc, bac̄, cab,c̄ba, bca and ac̄b. They can be obtained by exploitingthe group-subgroup relationship revealed in section3.5.2.1 of [18]G ⩽K (G)⩽ L(G)⩽NE (G), (C2)where K(G) is an intermediate group that retainsthe linear part of G and the translation part ofNE(G). L(G) =K(G)⊗{1, 1̄} if G does not haveinversion symmetry whileNE(G) is centrosymmetric,else L(G) =K(G). The group NE(G)/L(G) factorsout choices related to origin shift and inversion whichflips the handedness. The index nl = |NE(G)/L(G)|indicates the number of axes permutation types thatpreserves the CCS choice. For space groups, whennl = 6, the six different axis permutations do notinfluence the CCS choice. The order of the three basisvectors can be arbitrarily chosen. Spglib requires|ac|⩽ |bc|⩽ |cc|. Space groups P222, F222, Ibca, etcbelong to this case. Space group Pbca is the only onewith nl = 3, where abc, cab and bca preserves thestandard CCS choice, while bac̄, c̄ba and ac̄b changesthe choice to another. Spglib chooses ac to be theshortest vector, the orientation of bc and cc will besettled accordingly. nl = 2 means the standard choicefixes certain symmetry axis being parallel to cc. ac andbc cannot be determined by symmetry. Spglib addsthe constraint |ac|⩽ |bc|. Typical space groups areP2221,C222, etc. The rest orthorhombic space groupslike Pmc21 have nl = 1, of which the standard choicesuniquely determine the basis of the conventional cells.For layer groups, nl ⩽ 2 due to cc being the non-latticevector.For bulk, table 3.5.2.4 of [18] lists nl forcentrosymmetric NE(G), and nk = |NE(G)/K(G)|for noncentrosymmetric NE(G). For simplicity, weuse nl to denote the two symbols. Different metricparameter conditions may lead to different NE(G),and the one with highest symmetry should be con-sulted. For layer, nl can be deduced from the tableby consulting the line with highest symmetry underthe metric conditions a ̸= c and b ̸= c. It can also befound in [35].For space groups, additional correction matricesmust be applied for monoclinic and orthorhombicconventional cells with ‘A’, ‘B’ or body-centeringtypes, as the standard choices only possess ‘C’ center-ing lattices. The correction matrices are listed in [13].This step is not necessary for layer groups.Appendix D. Hall symbolsHall symbols are used to represent the differentCCS choices of space and layer groups in Spglib.Comparing to Hermann–Mauguin symbols, Hallsymbols take the origin of each symmetry opera-tion into consideration. A unique symmetry groupcan be deduced from a Hall symbol without addi-tional information. The 530 Hall symbols for dif-ferent choices of space groups are tabulated in tableA1.4.2.7 of [20]. As there does not exist a standardtable for layer group Hall symbols, we have producedtable 5 as a reference.The first column of the table gives the layer groupnumber. If there are multiple CCS choices for onelayer group type, the group number is followed bythe axis codes of the choice. The first row of eachgroup number always corresponds to the standardchoice defined in chapter 4 of [15]. The non-standardchoices are generated according to table 1.2.6.1 andchapter 4 of [15]. The second to the fourth columnlists the Hermann–Mauguin entries, Hall entries suit-able for computer processing and the Hall symbols,112D Mater. 11 (2024) 035009 J Fu et alTable 5. Table of the layer group numbers along with possible axis codes, Hermann–Mauguin symbols, Hall entries and Hall symbols.The table follows the format of table A1.4.2.7 in [20]. The first row of each layer group number demonstrates the standard choice. Allthe choices are compatible with the symmetry diagrams in chapter 4 and table 1.2.6.1 of [15]. The first letter ‘p’ and ‘c’ of the Hallsymbols are in lowercase to represent layer groups.n:c H–M entry Hall entry Hall symbol n:c H–M entry Hall entry Hall symbol1 p 1 p 1 p 1 34:b-ac p 2 a n p -2ab 2 p 2̄ab 22 p -1 -p 1 p̄ 1 35 c m 2 m c -2 -2 c 2̄ 2̄3:c p 1 1 2 p 2 p 2 35:b-ac c 2 m m c -2 2 c 2̄ 24:c p 1 1 m p -2 p 2̄ 36 c m 2 e c -2a -2a c 2̄a 2̄a5:c1 p 1 1 a p -2a p 2̄a 36:b-ac c 2 m e c -2a 2 c 2̄a 25:c2 p 1 1 n p -2ab p 2̄ab 37 p m mm -p 2 2 p̄ 2 25:c3 p 1 1 b p -2b p 2̄b 38 p m a a -p 2a 2 p̄ 2a 26:c p 1 1 2/m -p 2 p̄ 2 38:b-ac p b m b -p 2b 2b p̄ 2b 2b7:c1 p 1 1 2/a -p 2a p̄ 2a 39 p b a n -p 2ab 2b p̄ 2ab 2b7:c2 p 1 1 2/n -p 2ab p̄ 2ab 40 p m a m -p 2 2a p̄ 2 2a7:c3 p 1 1 2/b -p 2b p̄ 2b 40:b-ac p b m m -p 2 2b p̄ 2 2b8:a p 2 1 1 p 2x p 2x 41 p m m a -p 2a 2a p̄ 2a 2a8:b p 1 2 1 p 2y p 2y 41:b-ac p m m b -p 2b 2 p̄ 2b 29:a p 21 1 1 p 2xa p 2xa 42 p m a n -p 2ab 2 p̄ 2ab 29:b p 1 21 1 p 2yb p 2yb 42:b-ac p b m n -p 2ab 2ab p̄ 2ab 2ab10:a c 2 1 1 c 2x c 2x 43 p b a a -p 2a 2b p̄ 2a 2b10:b c 1 2 1 c 2y c 2y 43:b-ac p b a b -p 2b 2ab p̄ 2b 2ab11:a p m 1 1 p -2x p 2̄x 44 p b a m -p 2 2ab p̄ 2 2ab11:b p 1 m 1 p -2y p 2̄y 45 p b m a -p 2a 2ab p̄ 2a 2ab12:a p b 1 1 p -2xb p 2̄xb 45:b-ac p m a b -p 2b 2a p̄ 2b 2a12:b p 1 a 1 p -2ya p 2̄ya 46 p m m n -p 2ab 2a p̄ 2ab 2a13:a c m 1 1 c -2x c 2̄x 47 c m mm -c 2 2 c̄ 2 213:b c 1 m 1 c -2y c 2̄y 48 c m m e -c 2a 2 c̄ 2a 214:a p 2/m 1 1 -p 2x p̄ 2x 49 p 4 p 4 p 414:b p 1 2/m 1 -p 2y p̄ 2y 50 p -4 p -4 p 4̄15:a p 21/m 1 1 -p 2xa p̄ 2xa 51 p 4/m -p 4 p̄ 415:b p 1 21/m 1 -p 2yb p̄ 2yb 52:1 p 4/n:1 p 4 -1ab p 4 1̄ab16:a p 2/b 1 1 -p 2xb p̄ 2xb 52:2 p 4/n:2 -p 4a p̄ 4a16:b p 1 2/a 1 -p 2ya p̄ 2ya 53 p 4 2 2 p 4 2 p 4 217:a p 21/b 1 1 -p 2xab p̄ 2xab 54 p 4 21 2 p 4 2ab p 4 2ab17:b p 1 21/a 1 -p 2yab p̄ 2yab 55 p 4 m m p 4 -2 p 4 2̄18:a c 2/m 1 1 -c 2x c̄ 2x 56 p 4 b m p 4 -2ab p 4 2̄ab18:b c 1 2/m 1 -c 2y c̄ 2y 57 p -4 2 m p -4 2 p 4̄ 219 p 2 2 2 p 2 2 p 2 2 58 p -4 21 m p -4 2ab p 4̄ 2ab20 p 21 2 2 p 2 2a p 2 2a 59 p -4 m 2 p -4 -2 p 4̄ 2̄20:b-ac p 2 21 2 p 2 2b p 2 2b 60 p -4 b 2 p -4 -2ab p 4̄ 2̄ab21 p 21 21 2 p 2 2ab p 2 2ab 61 p 4/m mm -p 4 2 p̄ 4 222 c 2 2 2 c 2 2 c 2 2 62:1 p 4/n b m:1 p 4 2 -1ab p 4 2 1̄ab23 p m m 2 p 2 -2 p 2 2̄ 62:2 p 4/n b m:2 -p 4a 2b p̄ 4a 2b24 p m a 2 p 2 -2a p 2 2̄a 63 p 4/m b m -p 4 2ab p̄ 4 2ab24:b-ac p b m 2 p 2 -2b p 2 2̄b 64:1 p 4/n m m:1 p 4 2ab -1ab p 4 2ab 1̄ab25 p b a 2 p 2 -2ab p 2 2̄ab 64:2 p 4/n m m:2 -p 4a 2a p̄ 4a 2a26 c m m 2 c 2 -2 c 2 2̄ 65 p 3 p 3 p 327 p m 2 m p -2 -2 p 2̄ 2̄ 66 p -3 -p 3 p̄ 327:b-ac p 2 m m p -2 2 p 2̄ 2 67 p 3 1 2 p 3 2 p 3 228 p m 21 b p -2b -2 p 2̄b 2̄ 68 p 3 2 1 p 3 2” p 3 2”28:b-ac p 21 m a p -2a 2a p 2̄a 2a 69 p 3 m 1 p 3 -2” p 3 2̄”29 p b 21 m p -2 -2b p 2̄ 2̄b 70 p 3 1 m p 3 -2 p 3 2̄29:b-ac p 21 a m p -2 2a p 2̄ 2a 71 p -3 1 m -p 3 2 p̄ 3 230 p b 2 b p -2b -2b p 2̄b 2̄b 72 p -3 m 1 -p 3 2” p̄ 3 2”30:b-ac p 2 a a p -2a 2 p 2̄a 2 73 p 6 p 6 p 631 p m 2 a p -2a -2a p 2̄a 2̄a 74 p -6 p -6 p 6̄31:b-ac p 2 m b p -2b 2 p 2̄b 2 75 p 6/m -p 6 p̄ 632 p m 21 n p -2ab -2 p 2̄ab 2̄ 76 p 6 2 2 p 6 2 p 6 232:b-ac p 21 m n p -2ab 2ab p 2̄ab 2ab 77 p 6 m m p 6 -2 p 6 2̄33 p b 21 a p -2a -2ab p 2̄a 2̄ab 78 p -6 m 2 p -6 2 p 6̄ 233:b-ac p 21 a b p -2b 2a p 2̄b 2a 79 p -6 2 m p -6 -2 p 6̄ 2̄34 p b 2 n p -2ab -2ab p 2̄ab 2̄ab 80 p 6/m mm -p 6 2 p̄ 6 2122D Mater. 11 (2024) 035009 J Fu et alrespectively. The detailed meaning of the Hall sym-bols is described in [20], except we use symbols withthe first letter upper and lowercase to distinguish thespace and layer groups, respectively.ORCID iDsJingheng Fu https://orcid.org/0009-0002-6653-4155Kohei Shinohara https://orcid.org/0000-0002-5907-2549Kristian S Thygesen https://orcid.org/0000-0001-5197-214XReferences[1] Hammermesh M and Flammer C 1963 Group theory and itsapplication to physical problems Phys. Today 16 62[2] Wieder B J, Bradlyn B, Cano J, Wang Z, Vergniory M G,Elcoro L, Soluyanov A A, Felser C, Neupert T andRegnault N et al 2022 Topological materials discovery fromcrystal symmetry Nat. Rev. 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Introduction 2. Method 2.1. Primitive cell 2.2. Symmetry operations 2.3. Point group 2.4. Standardized conventional cell 2.5. Layer group 2.6. Symmetrization 3. Results 3.1. Layer group versus space group 4. Conclusions Appendix A. Classification of layer group A.1.  Point group of layer group A.2.  Translation lattice of layer group A.3.  Arithmetic-geometric crystal class A.4.  Layer group or layer group type? Appendix B. Delaunay reduction Appendix C. Conventional cells Appendix D. Hall symbols References