# Polyhedral group

Selected point groups in three dimensions :
| Involutional symmetry C$_s$, (*) [ ] = | Cyclic symmetry C$_{nv}$, (*nn) [n] = | Dihedral symmetry D$_{nh}$, (*n22) [n,2] = |
| Polyhedral group, [n,3], (*n32) |||
| Tetrahedral symmetry T$_d$, (*332) [3,3] = | Octahedral symmetry O$_h$, (*432) [4,3] = | Icosahedral symmetry I$_h$, (*532) [5,3] = |
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

## Groups

There are three polyhedral groups:
* The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. It is isomorphic to A$_4$.
** The conjugacy classes of T are:
*** identity
*** 4 × rotation by 120°, order 3, cw
*** 4 × rotation by 120°, order 3, ccw
*** 3 × rotation by 180°, order 2
* The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron. It is isomorphic to S$_4$.
** The conjugacy classes of O are:
*** identity
*** 6 × rotation by ±90° around vertices, order 4
*** 8 × rotation by ±120° around triangle centers, order 3
*** 3 × rotation by 180° around vertices, order 2
*** 6 × rotation by 180° around midpoints of edges, order 2
* The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron. It is isomorphic to A$_5$.
** The conjugacy classes of I are:
*** identity
*** 12 × rotation by ±72°, order 5
*** 12 × rotation by ±144°, order 5
*** 20 × rotation by ±120°, order 3
*** 15 × rotation by 180°, order 2
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 Dih$_2$, [2,2]. Pyritohedral symmetry is another doubling of tetrahedral symmetry.
The conjugacy classes of full tetrahedral symmetry, T$_d$ ≅ S$_4$, are:
* identity
* 8 × rotation by 120°
* 3 × rotation by 180°
* 6 × reflection in a plane through two rotation axes
* 6 × rotoreflection by 90°
The conjugacy classes of pyritohedral symmetry, T$_h$, include those of T, with the two classes of 4 combined, and each with inversion:
* identity
* 8 × rotation by 120°
* 3 × rotation by 180°
* inversion
* 8 × rotoreflection by 60°
* 3 × reflection in a plane
The conjugacy classes of the full octahedral group, O$_h$ ≅ S$_4$ × C$_2$, are:
* inversion
* 6 × rotoreflection by 90°
* 8 × rotoreflection by 60°
* 3 × reflection in a plane perpendicular to a 4-fold axis
* 6 × reflection in a plane perpendicular to a 2-fold axis
The conjugacy classes of full icosahedral symmetry, I$_h$ ≅ A$_5$ × C$_2$, include also each with inversion:
* inversion
* 12 × rotoreflection by 108°, order 10
* 12 × rotoreflection by 36°, order 10
* 20 × rotoreflection by 60°, order 6
* 15 × reflection, order 2

## Chiral polyhedral groups

Chiral polyhedral groups :
| Name (Orb.) | Coxeter notation | Order | Abstract structure | Rotation points #$_{valence}$ | Diagrams | Diagrams | Diagrams | Diagrams |
| Name (Orb.) | Coxeter notation | Order | Abstract structure | Rotation points #$_{valence}$ | Orthogonal | Stereographic | Stereographic | Stereographic |
| T (332) | [3,3]$^+$ | 12 | A$_4$ | 4$_3$ 3$_2$ | | | | |
| T$_h$ (3*2) | [4,3$^+$] | 24 | A$_4$ × C$_2$ | 4$_3$ 3$_{*2}$ | | | | |
| O (432) | [4,3]$^+$ | 24 | S$_4$ | 3$_4$ 4$_3$ 6$_2$ | | | | |
| I (532) | [5,3]$^+$ | 60 | A$_5$ | 6$_5$ 10$_3$ 15$_2$ | | | | |

## Full polyhedral groups

Full polyhedral groups :
| Weyl Schoe. (Orb.) | Coxeter notation | Order | Abstract structure | Coxeter number (h) | Mirrors (m) | Mirror diagrams | Mirror diagrams | Mirror diagrams | Mirror diagrams |
| Weyl Schoe. (Orb.) | Coxeter notation | Order | Abstract structure | Coxeter number (h) | Mirrors (m) | Orthogonal | Stereographic | Stereographic | Stereographic |
| A$_3$ T$_d$ (*332) | [3,3] | 24 | S$_4$ | 4 | 6 | | | | |
| B$_3$ O$_h$ (*432) | [4,3] | 48 | S$_4$ × C$_2$ | 8 | 3 >6 | | | | |
| H$_3$ I$_h$ (*532) | [5,3] | 120 | A$_5$ × C$_2$ | 10 | 15 | | | | |