Coxeter notation

Fundamental domains of reflective 3D point groups
, [ ] = [1]
C1v
, [2]
C2v
, [3]
C3v
, [4]
C4v
, [5]
C5v
, [6]
C6v

Order 2

Order 4

Order 6

Order 8

Order 10

Order 12

[2] = [2,1]
D1h

[2,2]
D2h

[2,3]
D3h

[2,4]
D4h

[2,5]
D5h

[2,6]
D6h

Order 4

Order 8

Order 12

Order 16

Order 20

Order 24
, [3,3], Td , [4,3], Oh , [5,3], Ih

Order 24

Order 48

Order 120
Coxeter notation expresses Coxeter groups as a list of branch orders of a Coxeter diagram, like the polyhedral groups, = [p,q]. Dihedral groups, , can be expressed as a product [ ]×[n] or in a single symbol with an explicit order 2 branch, [2,n].

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.