Categorisation of point groups by their unit cells

The various categorisations of classes of point groups can be arranged by the values taken of their unit cell lengths and angles, as shown in this diagram. The order of the group is defined as the number of symmetry operations. Follow the links to the character tables for each point group, or return to the top. Also the main point groups are listed with the common notations and symmetry operations.

Unit cell diagram

Conditions Point Groups Order Orthographic
projection

Triclinic abc
alpha not equal to beta gamma C₁ 1 Triclinic Orthographic Projection

C_i 2

Monoclinic abc
alpha = beta =/2 gamma C_s (C_1h) 2 Monoclinic Orthographic Projection

C₂ 2

C_2h 4

Orthorhombic abc
===/2 C_2v 4 Orthorhombic Orthographic Projection

D₂ 4

D_2h 8

Tetragonal a=bc
alpha = beta = gamma =/2 C₄ 4 Tetragona Orthographic Projection

S₄ 4

C_4h 8

D_2d 8

C_4v 8

D₄ 8

D_4h 16

Trigonal a=b=c
alpha = beta <2/3,
gamma not equal to /2 C₃ 3 Trigonal Orthographic Projection

S₆ 6

C_3v 6

D₃ 6

D_3d 12

Hexagonal a=bc
alpha = beta =/2,
gamma =2/3 C_3h 6 Hexagonal Orthographic Projection

C₆ 6

C_6h 12

D_3h 12

C_6v 12

D₆ 12

D_6h 24

Cubic* a=b=c
alpha = beta = gamma =/2 T 12 Cubic Orthographic Projection

T_h 24

T_d 24

O 24

O_h 48

	Conditions	Point Groups	Order	Orthographic projection
Triclinic	abc	C₁	1
Triclinic	abc	C_i	2

Monoclinic	abc ==/2	C_s (C_1h)	2
		C₂	2
		C_2h	4

Orthorhombic	abc ===/2	C_2v	4
		D₂	4
		D_2h	8

Tetragonal	a=bc ===/2	C₄	4
		S₄	4
		C_4h	8
		D_2d	8
		C_4v	8
		D₄	8
		D_4h	16

Trigonal	a=b=c =<2/3, /2	C₃	3
		S₆	6
		C_3v	6
		D₃	6
		D_3d	12

Hexagonal	a=bc ==/2, =2/3	C_3h	6
		C₆	6
		C_6h	12
		D_3h	12
		C_6v	12
		D₆	12
		D_6h	24

Cubic*	a=b=c ===/2	T	12
		T_h	24
		T_d	24
		O	24
		O_h	48

Categorisation of point groups by their unit cells

Last modified: Wed Dec 6 15:10:05 GMT 2000 by JPG (goss@excc.ex.ac.uk) V A

Last modified: Wed Dec 6 15:10:05 GMT 2000 by JPG (goss@excc.ex.ac.uk)
V A