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.

Cubic
Tetragonal
Orthorhombic
Monoclinic
Triclinic
Trigonal
Hexagonal
Operations Order Schönflies
Symbol
International
Symbol
Full
Symmetry
Symbol
Correlation
Table
Irred. Rep.
products
x i Isomorph.
with
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  
Non-Crystallographic
All