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
Operations Order Schönflies
Symbol
International
Symbol
Full
Symmetry
Symbol
Correlation
Table
Irred. Rep.
products
x i Isomorph.
with
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
Hexagonal
Non-Crystallographic
All