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Appendix a Space Groups

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Appendix a Space Groups Appendix A Space Groups Table A.1 gives the list of 230 three-periodic space groups. The point group inter- national symbols are underlined for the space groups that appear as the first ones in the list of a given crystal class. Table A.1 230 three-periodic space groups NG G IG NG G IG NG G IG Triclinic 1 1 1 C1 P1 2 Ci P1 Monoclinic 1 2 3 3 C2 P2 4 C2 P21 5 C2 C2 1 2 3 6 Cs Pm 7 Cs Pc 8 Cs Cm 4 1 / 2 / 9 Cs Cc 10 C2h P2 m 11 C2h P21 m 3 / 4 / 5 / 12 C2h C2 m 13 C2h P2 c 14 C2h P21 c 6 / 15 C2h C2 c Orthorhombic 1 2 3 16 D2 P222 17 D2 P2221 18 D2 P21212 4 5 6 19 D2 P212121 20 D2 C2221 21 D2 C222 7 8 9 22 D2 F222 23 D2 I 222 24 D2 I 212121 1 2 3 25 C2v Pmm2 26 C2v Pmc21 27 C2v Pcc2 4 5 6 28 C2v Pma2 29 C2v Pca21 30 C2v Pnc2 7 8 9 31 C2v Pmn21 32 C2v Pba2 33 C2v Pna21 10 11 12 34 C2v Pnn2 35 C2v Cmm2 36 C2v Cmc21 13 14 15 37 C2v Ccc2 38 C2v Amm2 39 C2v Abm2 16 17 18 40 C2v Ama2 41 C2v Aba2 42 C2v Fmm2 19 20 21 43 C2v Fdd2 44 C2v Imm2 45 C2v Iba2 22 1 2 46 C2v Ima2 47 D2h Pmmm 48 D2h Pnnn 3 4 5 49 D2h Pccm 50 D2h Pban 51 D2h Pmma (continued) © Springer-Verlag Berlin Heidelberg 2015 653 R.A. Evarestov, Theoretical Modeling of Inorganic Nanostructures, NanoScience and Technology, DOI 10.1007/978-3-662-44581-5 654 Appendix A: Space Groups Table A.1 (continued) NG G IG NG G IG NG G IG 6 7 8 52 D2h Pnna 53 D2h Pmna 54 D2h Pcca 9 10 11 55 D2h Pbam 56 D2h Pccn 57 D2h Pbcm 12 13 14 58 D2h Pnnm 59 D2h Pmmn 60 D2h Pbcn 15 16 17 61 D2h Pbca 62 D2h Pnma 63 D2h Cmcm 18 19 20 64 D2h Cmca 65 D2h Cmmm 66 D2h Cccm 21 22 23 67 D2h Cmma 68 D2h Ccca 69 D2h Fmmm 24 25 26 70 D2h Fddd 71 D2h Immm 72 D2h Ibam 27 28 73 D2h Ibca 74 D2h Imma Tetragonal 1 2 3 75 C4 P4 76 C4 P41 77 C4 P42 4 5 6 78 C4 P43 79 C4 I 4 80 C4 I 41 1 2 1 / 81 S4 P4 82 S4 I 4 83 C4h P4 m 2 / 3 / 4 / 84 C4h P42 m 85 C4h P4 n 86 C4h P42 n 5 / 6 / 1 87 C4h I 4 m 88 C4h I 41 a 89 D4 P422 2 3 4 90 D4 P4212 91 D4 P4122 92 D4 P41212 5 6 7 93 D4 P4222 94 D4 P42212 95 D4 P4322 8 9 10 96 D4 P43212 97 D4 I 422 98 D4 I 4122 1 2 3 99 C4v P4mm 100 C4v P4bm 101 C4v P42cm 4 5 6 102 C4v P42nm 103 C4v P4cc 104 C4v P4nc 7 8 9 105 C4v P42mc 106 C4v P42bc 107 C4v I 4mm 10 11 12 108 C4v I 4cm 109 C4v I 41md 110 C4v I 41cd 1 2 3 111 D2d P42m 112 D2d P42c 113 D2d P421m 4 5 6 114 D2d P421c 115 D2d P4m2 116 D2d P4c2 7 8 9 117 D2d P4b2 118 D2d P4n2 119 D2d I 4m2 10 11 12 120 D2d I 4c2 121 D2d I 42m 122 D2d I 42d 1 / 2 / 3 / 123 D4h P4 mmm 124 D4h P4 mcc 125 D4h P4 nbm 4 / 5 / 6 / 126 D4h P4 nnc 127 D4h P4 mbm 128 D4h P4 mnc 7 / 8 / 9 / 129 D4h P4 nmm 130 D4h P4 ncc 131 D4h P42 mmc 10 / 11 / 12 / 132 D4h P42 mcm 133 D4h P42 nbc 134 D4h P42 nnm 13 / 14 / 15 / 135 D4h P42 mbc 136 D4h P42 mnm 137 D4h P42 nmc 16 / 17 / 18 / 138 D4h P42 ncm 139 D4h I 4 mmm 140 D4h I 4 mcm 19 / 20 / 141 D4h I 41 amd 142 D4h I 41 acd Rhombohedral 1 2 3 143 C3 P3 144 C3 P31 145 C3 P32 4 1 2 146 C3 R3 147 C3i P3 148 C3i R3 1 2 3 149 D3 P312 150 D3 P321 151 D3 P3112 4 5 6 152 D3 P3121 153 D3 P3212 154 D3 P3221 7 1 2 155 D3 R32 156 C3v P3m1 157 C3v P31m (continued) Appendix A: Space Groups 655 Table A.1 (continued) NG G IG NG G IG NG G IG 3 4 5 158 C3v P3c1 159 C3v P31c 160 C3v R3m 6 1 2 161 C3v R3c 162 D3d P31m 163 D3d P31c 3 4 5 164 D3d P3m1 165 D3d P3c1 166 D3d R3m 6 167 D3d R3c Hexagonal 1 2 3 168 C6 P6 169 C6 P61 170 C6 P65 4 5 6 171 C6 P62 172 C6 P64 173 C6 P63 1 1 / 2 / 174 C3h P6 175 C6h P6 m 176 C6h P63 m 1 2 3 177 D6 P662 178 D6 P6122 179 D6 P6522 4 5 6 180 D6 P6222 181 D6 P6422 182 D6 P6322 1 2 3 183 C6v P6mm 184 C6v P6cc 185 C6v P63cm 4 1 2 186 C6v P63mc 187 D3h P6m2 188 D3h P6c2 3 4 1 / 189 D3h P62m 190 D3h P62c 191 D6h P6 mmm 2 / 3 / 4 / 192 D6h P6 mcc 193 D6h P63 mcm 194 D6h P63 mmc Cubic 195 T 1 P23 196 T 2 F23 197 T 3 I 23 4 5 1 198 T P213 199 T I 213 200 Th Pm3 2 3 4 201 Th Pn3 202 Th Fm3 203 Th Fd3 5 6 7 204 Th Im3 205 Th Pa3 206 Th Ia3 1 2 3 207 O P432 208 O P4232 209 O F432 4 5 6 210 O F4132 211 O I 432 212 O P4332 7 8 1 213 O P4132 214 O I 4132 215 Td P43m 2 3 4 216 Td F43m 217 Td I 43m 218 Td P43n 5 6 1 219 Td F43c 220 Td I 43d 221 Oh Pm3m 2 3 4 222 Oh Pn3n 223 Oh Pm3n 224 Oh Pn3m 5 6 7 225 Oh Fm3m 226 Oh Fm3c 227 Oh Fd3m 8 9 10 228 Oh Fd3c 229 Oh Im3m 230 Oh Ia3d Appendix B Layer Groups as Subroups of Space Groups Table B.1 shows the correspondence DG ↔ G/T3 between three-dimensional diperi- odic (DG) and three-periodic (G) space groups. This correspondence was at first given by Wood [1] in the setting that differs for rectangular diperiodic groups from that given in Table B.1 (the setting in this Table corresponds to the standard setting of [2]). Table B.2 presents the non-standard factorization of the diperiodic space groups. The generalized translational group, Z describes the periodical arrangement of the el- ementary motifs along two independent directions (these two directions are assumed to be in the xy-plane). Table B.1 Correspondence between three-dimensional two-periodic (DG) and three-periodic (G) space groups (DG ↔ G/T3), [2] DG G DG G DG G 1 2 1 1 P1 C1 1 28 Pm21b C2v 26 55 P4mm C4v 99 1 2 2 2 P1 Ci 2 29 Pb21m C2v 26 56 P4bm C4v 100 1 3 1 3 P112 C2 3 30 Pb2b C2v 27 57 P42m D2d 111 1 4 3 4 P11m Cs 6 31 Pm2a C2v 28 58 P421m D2d 113 2 7 5 5 P11a Cs 7 32 Pm21n C2v 31 59 P4m2 D2d 115 / 1 5 7 6 P112 m C2h 10 33 Pb21a C2v 29 60 P4b2 D2d 117 / 4 6 / 1 7 P112 a C2h 13 34 Pb2n C2v 30 61 P4 mmm D4h 123 1 14 / 3 8 P211 C2 3 35 Cm2m C2v 38 62 P4 nbm D4h 125 2 15 / 5 9 P2111 C2 4 36 Cm2a 4C2v 39 63 P4 mbm D4h 127 3 1 / 7 10 C211 C2 5 37 Pmmm D2h 47 64 P4 nmm D4h 129 1 3 1 11 Pm11 Cs 6 38 Pmaa D2h 49 65 P3 C3 143 2 4 1 12 Pb11 Cs 7 39 Pban D2h 50 66 P3 C3i 147 3 5 1 13 Cm11 Cs 8 40 Pmam D2h 51 67 P312 D3 149 / 1 5 2 14 P2 m11 C2h 10 41 Pmma 4D2h 51 68 P321 D3 150 (continued) © Springer-Verlag Berlin Heidelberg 2015 657 R.A. Evarestov, Theoretical Modeling of Inorganic Nanostructures, NanoScience and Technology, DOI 10.1007/978-3-662-44581-5 658 Appendix B: Layer Groups as Subroups of Space Groups Table B.1 (continued) DG G DG G DG G / 2 7 1 15 P21 m11 C2h 11 42 Pman D2h 53 69 P3m1 C3v 156 / 4 8 2 16 P2 b11 C2h 13 43 Pbaa D2h 54 70 P31m C3v 157 / 5 9 / 1 17 P21 b11 C2h 14 44 Pbam D2h 55 71 P312 m D3d 162 / 3 11 / 3 18 C2 m11 C2h 12 45 Pbma D2h 57 72 P32 m1 D3d 164 1 13 1 19 P222 D2 16 46 Pmmn D2h 59 73 P6 C6 168 2 19 1 20 P2122 D2 17 47 Cmmm D2h 65 74 P6 C3h 174 3 21 / 1 21 P21212 D2 18 48 Cmma D2h 67 75 P6 m C6h 175 6 1 1 22 C222 D2 21 49 P4 C4 75 76 P622 D6 177 1 1 1 23 Pmm2 C2v 25 50 P4 S4 81 77 P6mm C6v 183 4 / 1 1 24 Pma2 C2v 28 51 P4 m C4h 83 78 P6m2 D3h 187 8 / 3 3 25 Pba2 C2v 32 52 P4 n C4h 85 79 P62m D3h 189 11 1 / 1 26 Cmm2 C2v 35 53 P422 D4 89 80 P6 mmm D6h 191 1 2 27 Pm2m C2v 25 54 P4212 D4 90 The numbers of 17 plane groups are underlined Table B.2 The non-standard factorization of the diperiodic groups [3] DG H I PZ Int.symb.
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