Symmetry Pages

Categorisation by Unitcell

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
projection
Triclinic a≠b≠c
α≠β≠γ
C1 1 Triclinic Orthographic Projection
Ci 2
 
Monoclinic a≠b≠c
α=β=π/2≠γ
Cs(C1h) 2 Monoclinic Orthographic Projection
C2 2
C2h 4
 
Orthorhombic a≠b≠c
α=β=γ=π/2
C2v 4 Orthorhombic Orthographic Projection
D2 4
D2h 8
 
Tetragonal a="b"≠c
α=β=γ=π/2
C4 4 Tetragona Orthographic Projection
S4 4
C4h 8
D2d 8
C4v 8
D4 8
D4h 16
 
Trigonal a=b="c"
α=β<2π/3,
γ≠π/2
C3 3 Trigonal Orthographic Projection
S6 6
C3v 6
D3 6
D3d 12
 
Hexagonal a="b"≠c
α=β=π/2,
γ=2π/3
C3h 6 Hexagonal Orthographic Projection
C6 6
C6h 12
D3h 12
C6v 12
D6 12
D6h 24
 
Cubic * a=b="c"
α=β=γ=π/2
T 12 Cubic Orthographic Projection
Th 24
Td 24
O 24
Oh 48

Revised: © University of Newcastle upon Tyne, UK