Categorisation of point groups

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 T_h

E, 8C₃, 3C₂, 3 sigma _v, i, 8S₆ 24 T_h m3 T_h

E, 6C₄, 8C₃, 3C₂, 6C₂' 24 O 432 432 O_h T_d

E, 8C₃, 3C₂, 6S₄, 6 sigma _d 24 T_d 3m 3m O_h O

E, 8C₃, 6C₂, 6C₄, 3C₂', i, 6S₄, 8S₆, 3 sigma _h, 6 sigma _d 48 O_h m3m O_h

Tetragonal

Orthorhombic

Monoclinic

Triclinic

Trigonal

Hexagonal

Non-Crystallographic

All

Categorisation of point groups by symmetry operations

Last modified: Wed Dec 6 15:13:58 GMT 2000 by JPG (goss@excc.ex.ac.uk)
V A

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			T_h
E, 8C₃, 3C₂, 3_v, i, 8S₆	24	T_h	m3				T_h
E, 6C₄, 8C₃, 3C₂, 6C₂'	24	O	432	432			O_h	T_d
E, 8C₃, 3C₂, 6S₄, 6_d	24	T_d	3m	3m			O_h	O
E, 8C₃, 6C₂, 6C₄, 3C₂', i, 6S₄, 8S₆, 3_h, 6_d	48	O_h	m3m				O_h
Tetragonal
Orthorhombic
Monoclinic
Triclinic
Trigonal
Hexagonal
Non-Crystallographic
All

Categorisation of point groups by symmetry operations

Last modified: Wed Dec 6 15:13:58 GMT 2000 by JPG (goss@excc.ex.ac.uk) V A

Last modified: Wed Dec 6 15:13:58 GMT 2000 by JPG (goss@excc.ex.ac.uk)
V A