Space Group - Table of Space Groups in 3 Dimensions

Table of Space Groups in 3 Dimensions

# Crystal system Point group Space groups (international short symbol)
Intl Schönflies
1 Triclinic (2) 1 C1 Chiral P1
2 1 Ci P1
3–5 Monoclinic (13) 2 C2 Chiral P2, P21, C2
6–9 m Cs Pm, Pc, Cm, Cc
10–15 2/m C2h P2/m, P21/m, C2/m, P2/c, P21/c, C2/c
16–24 Orthorhombic (59) 222 D2 Chiral P222, P2221, P21212, P212121, C2221, C222, F222, I222, I212121
25–46 mm2 C2v Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2, Cmm2, Cmc21, Ccc2, Amm2, Aem2, Ama2, Aea2, Fmm2, Fdd2, Imm2, Iba2, Ima2
47–74 mmm D2h Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, Fmmm, Fddd, Immm, Ibam, Ibca, Imma
75–80 Tetragonal (68) 4 C4 Chiral P4, P41, P42, P43, I4, I41
81–82 4 S4 P4, I4
83–88 4/m C4h P4/m, P42/m, P4/n, P42/n, I4/m, I41/a
89–98 422 D4 Chiral P422, P4212, P4122, P41212, P4222, P42212, P4322, P43212, I422, I4122
99–110 4mm C4v P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc, I4mm, I4cm, I41md, I41cd
111–122 42m D2d P42m, P42c, P421m, P421c, P4m2, P4c2, P4b2, P4n2, I4m2, I4c2, I42m, I42d
123–142 4/mmm D4h P4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42/mmc, P42/mcm, P42/nbc, P42/nnm, P42/mbc, P42/mnm, P42/nmc, P42/ncm, I4/mmm, I4/mcm, I41/amd, I41/acd
143–146 Trigonal (25) 3 C3 Chiral P3, P31, P32, R3
147–148 3 S6 P3, R3
149–155 32 D3 Chiral P312, P321, P3112, P3121, P3212, P3221, R32
156–161 3m C3v P3m1, P31m, P3c1, P31c, R3m, R3c
162–167 3m D3d P31m, P31c, P3m1, P3c1, R3m, R3c,
168–173 Hexagonal (27) 6 C6 Chiral P6, P61, P65, P62, P64, P63
174 6 C3h P6
175–176 6/m C6h P6/m, P63/m
177–182 622 D6 Chiral P622, P6122, P6522, P6222, P6422, P6322
183–186 6mm C6v P6mm, P6cc, P63cm, P63mc
187–190 6m2 D3h P6m2, P6c2, P62m, P62c
191–194 6/mmm D6h P6/mmm, P6/mcc, P63/mcm, P63/mmc
195–199 Cubic (36) 23 T Chiral P23, F23, I23, P213, I213
200–206 m3 Th Pm3, Pn3, Fm3, Fd3, Im3, Pa3, Ia3
207–214 432 O Chiral P432, P4232, F432, F4132, I432, P4332, P4132, I4132
215–220 43m Td P43m, F43m, I43m, P43n, F43c, I43d
221–230 m3m Oh Pm3m, Pn3n, Pm3n, Pn3m, Fm3m, Fm3c, Fd3m, Fd3c, Im3m, Ia3d

Note. An e plane is a double glide plane, one having glides in two different directions. They are found in seven orthorombic, five tetragonal and five cubic space groups, all with centered lattice. The use of the symbol e became official with Hahn (2002).

The lattice system can be found as follows. If the crystal system is not trigonal then the lattice system is of the same type. If the crystal system is trigonal, then the lattice system is hexagonal unless the space group is one of the seven in the rhombohedral lattice system consisting of the 7 trigonal space groups in the table above whose name begins with R. (The term rhombohedral system is also sometimes used as an alternative name for the whole trigonal system.) The hexagonal lattice system is larger than the hexagonal crystal system, and consists of the hexagonal crystal system together with the 18 groups of the trigonal crystal system other than the seven whose names begin with R.

The Bravais lattice of the space group is determined by the lattice system together with the initial letter of its name, which for the non-rhombohedral groups is P, I, F, or C, standing for the principal, body centered, face centered, or C-face centered lattices.

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