University of Exeter

Categorisation of point groups by symmetry operations

Exeter Symmetry Pages

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. maps to
symbol links to a correlation table, and Irreducible representation product 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
Irred. Rep.
x i Isomorph.
E, 4C3, 4C32, 3C2 12 T 23 23   product Th  
E, 8C3, 3C2, 3sigmav, i, 8S6 24 Th m3 2/m-3   product Th  
E, 6C4, 8C3, 3C2, 6C2' 24 O 432 432   product Oh Td
E, 8C3, 3C2, 6S4, 6sigmad 24 Td -43m -43m correlation product Oh O
E, 8C3, 6C2, 6C4, 3C2', i, 6S4, 8S6, 3sigmah, 6sigmad 48 Oh m3m 4/m-32/m correlation product Oh  
E, C4, C2, C43 4 C4 4 4   C4 lvms C4h S4
E, S4, C2, S43 4 S4 -4 -4 correlation S4 lvms C4h C4
E, C4, C2, C43, i, S43, sigmah, S4 8 C4h 4/m 4/m correlation product C4h  
E, 2C4, C2, 2C2', 2C2'' 8 D4 422 422   product D4h C4v, D2d
E, 2C4, C2, 2sigmav, 2sigmad 8 C4v 4mm 4mm correlation product D4h D4, D2d
E, 2S4, C2, 2C2', 2sigmad 8 D2d (Vd) -42m -42m correlation product D4h D4, C4v
E, 2C4, C2, 2C2', 2C2'', i, 2S4, sigmah, 2sigmav, 2sigmad 16 D4h 4/mmm 4/m4/m4/m correlation product D4h  
E, C2, C2', C2'' 4 D2 (V) 222 222   product D2h C2v, C2h
E, C2, sigmav, sigmav' 4 C2v mm2 mm2 correlation product D2h D2, C2h
E, C2, C2', C2'', i, sigma, sigma', sigma'' 8 D2h (Vh) mmm 2/m2/m2/m correlation product 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, sigmav and sigmav' coincide with the planes (1-10) and (001)). Also belonging to type I are centres for which C2 coincides with [001] and sigmav and sigmav' with (110) and (1-10).
Type II: C2 axis coincides with [001] and the axes C2' and C2'' with [100] and [010], (or, alternatively, sigmav and sigmav' coincide with (010) and (100)).
E, C2 2 C2 2 2   product C2h Cs, Ci
E, sigmah 2 Cs (C1h) m m   product C2h C2, Ci
E, C2, i, sigmah 4 C2h 2/m 2/m correlation product C2h D2, C2v
E 1 C1 1 1   product Ci
E, i 2 Ci (S2) -1 -1   product Ci Cs, C2
E, C3, C32 3 C3 3 3   product S6  
E, C3, C32, i, S65, S6 6 S6 (C3i) -3 -3 correlation product S6 C6, C3h
E, 2C3, 3C2 6 D3 32 32   product D3d C3v
E, 2C3, 3sigmav 6 C3v 3m 3m correlation product D3d D3
E, 2C3, 3C2, i, 2S6, 3sigmad 12 D3d -3m -32/m correlation product D3d C6v, D6, D3h
E, C6, C3, C2, C32, C65 6 C6 6 6   product C6h S6, C3h
E, C3, C32, sigmah, S3, S32 6 C3h (S3) -6 -6 correlation product C6h S6, C6
E, C6, C3, C2, C32, C65, i, S32, S65, sigmah, S6, S3 12 C6h 6/m 6/m correlation product C6h  
E, 2C6, 2C3, C2, 3C2', 3C2'' 12 D6 622 622   product D6h C6v, D3d, D3h
E, 2C6, 2C3, C2, 3sigmav, 3sigmad 12 C6v 6mm 6mm correlation product D6h D6, D3d, D3h
E, 2C3, 3C2, sigmah, 2S3, 3sigmav 12 D3h -6m2-6m2 correlation product D6h D6, D3d, C6v
E, 2C6, 2C5, C2, 3C2', 3C2'', i, 2S3, 2S6, sigmah, 3sigmad, 3sigmav 24 D6h 6/mmm 6/m2/m2/m correlation product D6h  
Operations Order Schönflies
Irred. Rep.
x i Isomorph.
E, 2C infinity infinity Cinfinity infinity -   - Cinfinityh  
E, 2C infinity, i, 2S infinity infinity Cinfinityh infinity/m -   - Cinfinityh  
E, 2C infinity, infinitysigmav infinity Cinfinityv infinitym -   - Dinfinityh  
E, 2C infinity, infinitysigmav, i, 2S infinity, infinityC2 infinity Dinfinityh infinity/mm -   - Dinfinityh  
E, C5, C52, C53,  C54 5 C5 5 -   product S10  
E, S8, C4, S83, C2, S85, C43, S87 8 S8 - - correlation product C8h  
E, 2C5, 2C52, 5C2 10 D5 - -   product D5d C5v
E, 2C5, 2C52, 5sigmav 10 C5v - - correlation product D5d D5
E, C5, C52, C53, C54, sigmah, S5, S57, S53, S59 10 C5h - - correlation product C10h  
E, 2S8, 2C4, 2S83, C2, 4C2', 4sigmad 16 D4d - - correlation product D8h  
E, 2C5, 2C52, 5C2, i, 2S10, 2S103, 5sigmad 20 D5d - - correlation product D5d D5h
E, 2C5, 2C52, 5C2, sigmah, 2S5, 2S52, 5sigmad 20 D5h - - correlation product D10h D5d
E 2S12 2C6 2S4 2C3 2S125 C2 6C2' 6sigmad 24 D6d - - correlation product D12h  
E, 12C5, 12C52, 20C3, 15C2 60 I - -   product Ih  
E, 12C5, 12C52, 20C3, 15C2, i, 12S10, 12S103, 20S6, 15sigma 120 Ih - -   product Ih