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