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