# Categorisation of point groups by symmetry operations

This table lists point group symmetries along with their symmetry operations, the order of the group (i.e. the number of symmetry operations) and common notations. links to a correlation table, and links to tables of products of irreducible representations. The group produced by combination with inversion is listed under "x i". This, in the case of crystolographic point groups, is the Laue class which corresponds to the symmetry of reciprocal space. Isomorphic groups are also listed where character tables are available.

 Operations Order SchönfliesSymbol InternationalSymbol FullSymmetrySymbol CorrelationTable Irred. Rep.products x i Isomorph.with Cubic E, 4C3, 4C32, 3C2 12 T 23 23 Th E, 8C3, 3C2, 3v, i, 8S6 24 Th m3 Th E, 6C4, 8C3, 3C2, 6C2' 24 O 432 432 Oh Td E, 8C3, 3C2, 6S4, 6d 24 Td 3m 3m Oh O E, 8C3, 6C2, 6C4, 3C2', i, 6S4, 8S6, 3h, 6d 48 Oh m3m Oh Tetragonal E, C4, C2, C43 4 C4 4 4 C4h S4 E, S4, C2, S43 4 S4 C4h C4 E, C4, C2, C43, i, S43, h, S4 8 C4h 4/m C4h E, 2C4, C2, 2C2', 2C2'' 8 D4 422 422 D4h C4v, D2d E, 2C4, C2, 2v, 2d 8 C4v 4mm 4mm D4h D4, D2d E, 2S4, C2, 2C2', 2d 8 D2d (Vd) 2m 2m D4h D4, C4v E, 2C4, C2, 2C2', 2C2'', i, 2S4, h, 2v, 2d 16 D4h 4/mmm D4h Orthorhombic E, C2, C2', C2'' 4 D2 (V) 222 222 D2h C2v, C2h E, C2, v, v' 4 C2v mm2 mm2 D2h D2, C2h E, C2, C2', C2'', i, , ', '' 8 D2h (Vh) mmm D2h Rhombic symmetry for defects in crystals is often divided into two types: Type I: C2 coincides with the [110] direction and C2' and C2'' with [001] and [1-10] directions respectively (or, v and v' coincide with the planes (1-10) and (001)). Also belonging to type I are centres for which C2 coincides with [001] and v and v' with (110) and (1-10). Type II: C2 axis coincides with [001] and the axes C2' and C2'' with [100] and [010], (or, alternatively, v and v' coincide with (010) and (100)). Monoclinic E, C2 2 C2 2 2 C2h Cs, Ci E, h 2 Cs (C1h) m m C2h C2, Ci E, C2, i, h 4 C2h 2/m C2h D2, C2v Triclinic E 1 C1 1 1 Ci E, i 2 Ci (S2) Ci Cs, C2 Trigonal E, C3, C32 3 C3 3 3 S6 E, C3, C32, i, S65, S6 6 S6 (C3i) S6 C6, C3h E, 2C3, 3C2 6 D3 32 32 D3d C3v E, 2C3, 3v 6 C3v 3m 3m D3d D3 E, 2C3, 3C2, i, 2S6, 3d 12 D3d m D3d C6v, D6, D3h Hexagonal E, C6, C3, C2, C32, C65 6 C6 6 6 C6h S6, C3h E, C3, C32, h, S3, S32 6 C3h (S3) C6h S6, C6 E, C6, C3, C2, C32, C65, i, S32, S65, h, S6, S3 12 C6h 6/m C6h E, 2C6, 2C3, C2, 3C2', 3C2'' 12 D6 622 622 D6h C6v, D3d, D3h E, 2C6, 2C3, C2, 3v, 3d 12 C6v 6mm 6mm D6h D6, D3d, D3h E, 2C3, 3C2, h, 2S3, 3v 12 D3h m2 m2 D6h D6, D3d, C6v E, 2C6, 2C5, C2, 3C2', 3C2'', i, 2S3, 2S6, h, 3d, 3v 24 D6h 6/mmm D6h Non-Crystallographic Operations Order SchönfliesSymbol InternationalSymbol FullSymmetrySymbol CorrelationTable Irred. Rep.products x i Isomorph.with E, 2 C - - Ch E, 2, i, 2 Ch /m - - Ch E, 2, v Cv m - - Dh E, 2, v, i, 2, C2 Dh /mm - - Dh E, C5, C52, C53,  C54 5 C5 5 - S10 E, S8, C4, S83, C2, S85, C43, S87 8 S8 - - C8h E, 2C5, 2C52, 5C2 10 D5 - - D5d C5v E, 2C5, 2C52, 5v 10 C5v - - D5d D5 E, C5, C52, C53, C54, h, S5, S57, S53, S59 10 C5h - - C10h E, 2S8, 2C4, 2S83, C2, 4C2', 4d 16 D4d - - D8h E, 2C5, 2C52, 5C2, i, 2S10, 2S103, 5d 20 D5d - - D5d D5h E, 2C5, 2C52, 5C2, h, 2S5, 2S52, 5d 20 D5h - - D10h D5d E 2S12 2C6 2S4 2C3 2S125 C2 6C2' 6d 24 D6d - - D12h E, 12C5, 12C52, 20C3, 15C2 60 I - - Ih E, 12C5, 12C52, 20C3, 15C2, i, 12S10, 12S103, 20S6, 15 120 Ih - - Ih