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 | Schönflies Symbol |
International Symbol |
Full Symmetry Symbol |
Correlation Table |
Irred. Rep. products |
x i | Isomorph. with |
Cubic | ||||||||
E, 4C_{3}, 4C_{3}^{2}, 3C_{2} | 12 | T | 23 | 23 | T_{h} | |||
E, 8C_{3}, 3C_{2}, 3_{v}, i, 8S_{6} | 24 | T_{h} | m3 | T_{h} | ||||
E, 6C_{4}, 8C_{3}, 3C_{2}, 6C_{2}' | 24 | O | 432 | 432 | O_{h} | T_{d} | ||
E, 8C_{3}, 3C_{2}, 6S_{4}, 6_{d} | 24 | T_{d} | 3m | 3m | O_{h} | O | ||
E, 8C_{3}, 6C_{2}, 6C_{4}, 3C_{2}', i, 6S_{4}, 8S_{6}, 3_{h}, 6_{d} | 48 | O_{h} | m3m | O_{h} | ||||
Tetragonal | ||||||||
E, C_{4}, C_{2}, C_{4}^{3} | 4 | C_{4} | 4 | 4 | C_{4h} | S_{4} | ||
E, S_{4}, C_{2}, S_{4}^{3} | 4 | S_{4} | C_{4h} | C_{4} | ||||
E, C_{4}, C_{2}, C_{4}^{3}, i, S_{4}^{3}, _{h}, S_{4} | 8 | C_{4h} | 4/m | C_{4h} | ||||
E, 2C_{4}, C_{2}, 2C_{2}', 2C_{2}'' | 8 | D_{4} | 422 | 422 | D_{4h} | C_{4v}, D_{2d} | ||
E, 2C_{4}, C_{2}, 2_{v}, 2_{d} | 8 | C_{4v} | 4mm | 4mm | D_{4h} | D_{4}, D_{2d} | ||
E, 2S_{4}, C_{2}, 2C_{2}', 2_{d} | 8 | D_{2d} (V_{d}) | 2m | 2m | D_{4h} | D_{4}, C_{4v} | ||
E, 2C_{4}, C_{2}, 2C_{2}', 2C_{2}'', i, 2S_{4}, _{h}, 2_{v}, 2_{d} | 16 | D_{4h} | 4/mmm | D_{4h} | ||||
Orthorhombic | ||||||||
E, C_{2}, C_{2}', C_{2}'' | 4 | D_{2} (V) | 222 | 222 | D_{2h} | C_{2v}, C_{2h} | ||
E, C_{2}, _{v}, _{v}' | 4 | C_{2v} | mm2 | mm2 | D_{2h} | D_{2}, C_{2h} | ||
E, C_{2}, C_{2}', C_{2}'', i, , ', '' | 8 | D_{2h} (V_{h}) | mmm | D_{2h} | ||||
Rhombic symmetry for defects in crystals is often divided into two
types: Type I: C_{2} coincides with the [110] direction and C_{2}' and C_{2}'' 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 C_{2} coincides with [001] and _{v} and _{v}' with (110) and (1-10). Type II: C_{2} axis coincides with [001] and the axes C_{2}' and C_{2}'' with [100] and [010], (or, alternatively, _{v} and _{v}' coincide with (010) and (100)). |
||||||||
Monoclinic | ||||||||
E, C_{2} | 2 | C_{2} | 2 | 2 | C_{2h} | C_{s}, C_{i} | ||
E, _{h} | 2 | C_{s} (C_{1h}) | m | m | C_{2h} | C_{2}, C_{i} | ||
E, C_{2}, i, _{h} | 4 | C_{2h} | 2/m | C_{2h} | D_{2}, C_{2v} | |||
Triclinic | ||||||||
E | 1 | C_{1} | 1 | 1 | C_{i} | |||
E, i | 2 | C_{i} (S_{2}) | C_{i} | C_{s}, C_{2} | ||||
Trigonal | ||||||||
E, C_{3}, C_{3}^{2} | 3 | C_{3} | 3 | 3 | S_{6} | |||
E, C_{3}, C_{3}^{2}, i, S_{6}^{5}, S_{6} | 6 | S_{6} (C_{3i}) | S_{6} | C_{6}, C_{3h} | ||||
E, 2C_{3}, 3C_{2} | 6 | D_{3} | 32 | 32 | D_{3d} | C_{3v} | ||
E, 2C_{3}, 3_{v} | 6 | C_{3v} | 3m | 3m | D_{3d} | D_{3} | ||
E, 2C_{3}, 3C_{2}, i, 2S_{6}, 3_{d} | 12 | D_{3d} | m | D_{3d} | C_{6v}, D_{6}, D_{3h} | |||
Hexagonal | ||||||||
E, C_{6}, C_{3}, C_{2}, C_{3}^{2}, C_{6}^{5} | 6 | C_{6} | 6 | 6 | C_{6h} | S_{6}, C_{3h} | ||
E, C_{3}, C_{3}^{2}, _{h}, S_{3}, S_{3}^{2} | 6 | C_{3h} (S_{3}) | C_{6h} | S_{6}, C_{6} | ||||
E, C_{6}, C_{3}, C_{2}, C_{3}^{2}, C_{6}^{5}, i, S_{3}^{2}, S_{6}^{5}, _{h}, S_{6}, S_{3} | 12 | C_{6h} | 6/m | C_{6h} | ||||
E, 2C_{6}, 2C_{3}, C_{2}, 3C_{2}', 3C_{2}'' | 12 | D_{6} | 622 | 622 | D_{6h} | C_{6v}, D_{3d}, D_{3h} | ||
E, 2C_{6}, 2C_{3}, C_{2}, 3_{v}, 3_{d} | 12 | C_{6v} | 6mm | 6mm | D_{6h} | D_{6}, D_{3d}, D_{3h} | ||
E, 2C_{3}, 3C_{2}, _{h}, 2S_{3}, 3_{v} | 12 | D_{3h} | m2 | m2 | D_{6h} | D_{6}, D_{3d}, C_{6v} | ||
E, 2C_{6}, 2C_{5}, C_{2}, 3C_{2}', 3C_{2}'', i, 2S_{3}, 2S_{6}, _{h}, 3_{d}, 3_{v} | 24 | D_{6h} | 6/mmm | D_{6h} | ||||
Non-Crystallographic | ||||||||
Operations | Order | Schönflies Symbol |
International Symbol |
Full Symmetry Symbol |
Correlation Table |
Irred. Rep. products |
x i | Isomorph. with |
E, 2 | C_{} | - | - | C_{h} | ||||
E, 2, i, 2 | C_{h} | /m | - | - | C_{h} | |||
E, 2, _{v} | C_{v} | m | - | - | D_{h} | |||
E, 2, _{v}, i, 2, C_{2} | D_{h} | /mm | - | - | D_{h} | |||
E, C_{5}, C_{5}^{2}, C_{5}^{3}, C_{5}^{4} | 5 | C_{5} | 5 | - | S_{10} | |||
E, S_{8}, C_{4}, S_{8}^{3}, C_{2}, S_{8}^{5}, C_{4}^{3}, S_{8}^{7} | 8 | S_{8} | - | - | C_{8h} | |||
E, 2C_{5}, 2C_{5}^{2}, 5C_{2} | 10 | D_{5} | - | - | D_{5d} | C_{5v} | ||
E, 2C_{5}, 2C_{5}^{2}, 5_{v} | 10 | C_{5v} | - | - | D_{5d} | D_{5} | ||
E, C_{5}, C_{5}^{2}, C_{5}^{3}, C_{5}^{4}, _{h}, S_{5}, S_{5}^{7}, S_{5}^{3}, S_{5}^{9} | 10 | C_{5h} | - | - | C_{10h} | |||
E, 2S_{8}, 2C_{4}, 2S_{8}^{3}, C_{2}, 4C_{2}', 4_{d} | 16 | D_{4d} | - | - | D_{8h} | |||
E, 2C_{5}, 2C_{5}^{2}, 5C_{2}, i, 2S_{10}, 2S_{10}^{3}, 5_{d} | 20 | D_{5d} | - | - | D_{5d} | D_{5h} | ||
E, 2C_{5}, 2C_{5}^{2}, 5C_{2}, _{h}, 2S_{5}, 2S_{5}^{2}, 5_{d} | 20 | D_{5h} | - | - | D_{10h} | D_{5d} | ||
E 2S_{12} 2C_{6} 2S_{4} 2C_{3} 2S_{12}^{5} C_{2} 6C_{2}' 6_{d} | 24 | D_{6d} | - | - | D_{12h} | |||
E, 12C_{5}, 12C_{5}^{2}, 20C_{3}, 15C_{2} | 60 | I | - | - | I_{h} | |||
E, 12C_{5}, 12C_{5}^{2}, 20C_{3}, 15C_{2}, i, 12S_{10}, 12S_{10}^{3}, 20S_{6}, 15 | 120 | I_{h} | - | - | I_{h} |