Polyhedral group

Point groups in three dimensions

Involutional symmetry
Cs, (*)
[ ] =

Cyclic symmetry
Cnv, (*nn)
[n] =

Dihedral symmetry
Dnh, (*n22)
[n,2] =
Polyhedral group, [n,3], (*n32)

Tetrahedral symmetry
Td, (*332)
[3,3] =

Octahedral symmetry
Oh, (*432)
[4,3] =

Icosahedral symmetry
Ih, (*532)
[5,3] =

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

Groups

There are three polyhedral groups:

These symmetries double to 24, 48, 120 respectively for the full reflectional groups. The reflection symmetries have 6, 9, and 15 mirrors respectively. The octahedral symmetry, [4,3] can be seen as the union of 6 tetrahedral symmetry [3,3] mirrors, and 3 mirrors of dihedral symmetry Dih2, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry.

The conjugacy classes of full tetrahedral symmetry, Td, are:

The conjugacy classes of pyritohedral symmetry, Th, include those of T, with the two classes of 4 combined, and each with inversion:

The conjugacy classes of the full octahedral group, Oh, are:

The conjugacy classes of full icosahedral symmetry Ih include also each with inversion:

Chiral polyhedral groups

Chiral polyhedral groups
Name
(Orb.)
Coxeter
notation
Order Abstract
structure
Rotation
points
#valence
Diagrams
Orthogonal Stereographic
T
(332)

[3,3]+
12A443
32
Th
(3*2)


[4,3+]
24A4×2 43
3*2
O
(432)

[4,3]+
24S4 34
43
62
I
(532)

[5,3]+
60A565
103
152

Full polyhedral groups

Full polyhedral groups
Weyl
Schoe.
(Orb.)
Coxeter
notation
Order Abstract
structure
Coxeter
number

(h)
Mirrors
(m)
Mirror diagrams
Orthogonal Stereographic
A3
Td
(*332)


[3,3]
24 S4 4 6
B3
Oh
(*432)


[4,3]
48 S4×28 3
6
H3
Ih
(*532)


[5,3]
120 A5×210 15

See also

References

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