University of Exeter

Categorisation of point groups by their unit cells

Exeter Symmetry Pages

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
Triclinic anot equal tobnot equal toc
alphanot equal tobetanot equal togamma
C1 1 Triclinic Orthographic Projection
Ci 2
Monoclinic anot equal tobnot equal toc
alpha=beta=pi/2not equal togamma
Cs (C1h) 2 Monoclinic Orthographic Projection
C2 2
C2h 4
Orthorhombic anot equal tobnot equal toc
C2v 4 Orthorhombic Orthographic Projection
D2 4
D2h 8
Tetragonal a=bnot equal toc
C4 4 Tetragona Orthographic Projection
S4 4
C4h 8
D2d 8
C4v 8
D4 8
D4h 16
Trigonal a=b=c
gammanot equal topi/2
C3 3 Trigonal Orthographic Projection
S6 6
C3v 6
D3 6
D3d 12
Hexagonal a=bnot equal toc
C3h 6 Hexagonal Orthographic Projection
C6 6
C6h 12
D3h 12
C6v 12
D6 12
D6h 24
Cubic* a=b=c
T 12 Cubic Orthographic Projection
Th 24
Td 24
O 24
Oh 48