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

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, 2 C infinity C - - C_h

E, 2, i, 2 S infinity C_h /m - - C_h

E, 2, sigma _v C_v m - - D_h

E, 2 C infinity , sigma _v, i, 2, C₂ D_h /mm - - D_h

E, C₅, C₅², C₅³, C₅⁴ 5 C₅ 5 - S₁₀

E, S₈, C₄, S₈³, C₂, S₈⁵, C₄³, S₈⁷ 8 S₈ - - C_8h

E, 2C₅, 2C₅², 5C₂ 10 D₅ - - D_5d C_5v

E, 2C₅, 2C₅², 5 sigma _v 10 C_5v - - D_5d D₅

E, C₅, C₅², C₅³, C₅⁴, sigma _h, S₅, S₅⁷, S₅³, S₅⁹ 10 C_5h - - C_10h

E, 2S₈, 2C₄, 2S₈³, C₂, 4C₂', 4 sigma _d 16 D_4d - - D_8h

E, 2C₅, 2C₅², 5C₂, i, 2S₁₀, 2S₁₀³, 5 sigma _d 20 D_5d - - D_5d D_5h

E, 2C₅, 2C₅², 5C₂, sigma _h, 2S₅, 2S₅², 5 sigma _d 20 D_5h - - D_10h D_5d

E 2S₁₂ 2C₆ 2S₄ 2C₃ 2S₁₂⁵ C₂ 6C₂' 6 sigma _d 24 D_6d - - D_12h

E, 12C₅, 12C₅², 20C₃, 15C₂ 60 I - - I_h

E, 12C₅, 12C₅², 20C₃, 15C₂, i, 12S₁₀, 12S₁₀³, 20S₆, 15 sigma 120 I_h - - I_h

All

Categorisation of point groups by symmetry operations

Last modified: Thu Nov 23 12:00:11 GMT 2000 by JPG (goss@excc.ex.ac.uk)
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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, 2		C		-		-	C_h
E, 2, i, 2		C_h	/m	-		-	C_h
E, 2, _v		C_v	m	-		-	D_h
E, 2, _v, i, 2, C₂		D_h	/mm	-		-	D_h
E, C₅, C₅², C₅³, C₅⁴	5	C₅	5	-			S₁₀
E, S₈, C₄, S₈³, C₂, S₈⁵, C₄³, S₈⁷	8	S₈	-	-			C_8h
E, 2C₅, 2C₅², 5C₂	10	D₅	-	-			D_5d	C_5v
E, 2C₅, 2C₅², 5_v	10	C_5v	-	-			D_5d	D₅
E, C₅, C₅², C₅³, C₅⁴, _h, S₅, S₅⁷, S₅³, S₅⁹	10	C_5h	-	-			C_10h
E, 2S₈, 2C₄, 2S₈³, C₂, 4C₂', 4_d	16	D_4d	-	-			D_8h
E, 2C₅, 2C₅², 5C₂, i, 2S₁₀, 2S₁₀³, 5_d	20	D_5d	-	-			D_5d	D_5h
E, 2C₅, 2C₅², 5C₂, _h, 2S₅, 2S₅², 5_d	20	D_5h	-	-			D_10h	D_5d
E 2S₁₂ 2C₆ 2S₄ 2C₃ 2S₁₂⁵ C₂ 6C₂' 6_d	24	D_6d	-	-			D_12h
E, 12C₅, 12C₅², 20C₃, 15C₂	60	I	-	-			I_h
E, 12C₅, 12C₅², 20C₃, 15C₂, i, 12S₁₀, 12S₁₀³, 20S₆, 15	120	I_h	-	-			I_h
All

Categorisation of point groups by symmetry operations

Last modified: Thu Nov 23 12:00:11 GMT 2000 by JPG (goss@excc.ex.ac.uk) V A

Last modified: Thu Nov 23 12:00:11 GMT 2000 by JPG (goss@excc.ex.ac.uk)
V A