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