Categorisation of point groups by symmetry operations

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 S₆

E, C₃, C₃², i, S₆⁵, S₆ 6 S₆ (C_3i) S₆ C₆, C_3h

E, 2C₃, 3C₂ 6 D₃ 32 32 D_3d C_3v

E, 2C₃, 3 sigma _v 6 C_3v 3m 3m D_3d D₃

E, 2C₃, 3C₂, i, 2S₆, 3 sigma _d 12 D_3d m D_3d C_6v, D₆, D_3h

Hexagonal

Non-Crystallographic

All

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			S₆
E, C₃, C₃², i, S₆⁵, S₆	6	S₆ (C_3i)					S₆	C₆, C_3h
E, 2C₃, 3C₂	6	D₃	32	32			D_3d	C_3v
E, 2C₃, 3_v	6	C_3v	3m	3m			D_3d	D₃
E, 2C₃, 3C₂, i, 2S₆, 3_d	12	D_3d	m				D_3d	C_6v, D₆, D_3h
Hexagonal
Non-Crystallographic
All

Categorisation of point groups by symmetry operations

Last modified: Wed Nov 22 17:17:53 GMT 2000 by JPG (goss@excc.ex.ac.uk) V A

Last modified: Wed Nov 22 17:17:53 GMT 2000 by JPG (goss@excc.ex.ac.uk)
V A