 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.
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 ≅ S4 , 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 ≅ S4 × C2 , are:
The conjugacy classes of full icosahedral symmetry, Ih ≅ A5 × C2 , include also each with inversion:
| Name (Orb.) | Coxeter notation | Order | Abstract structure | Rotation points # valence | Diagrams | |||
|---|---|---|---|---|---|---|---|---|
| Orthogonal | Stereographic | |||||||
| T (332) |  [3,3]+ | 12 | A4 | 43   32  |  |  |  |  |
| Th (3*2) |  [4,3+] | 24 | A4 × C2 | 43 3*2 |  |  |  |  |
| O (432) | [4,3]+ | 24 | S4 | 34  43 62 |  |  |  |  |
| I (532) | [5,3]+ | 60 | A5 | 65  103 152 |  |  |  |  |
| 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 × C2 | 8 | node_c2} >6 |  |  |  |  |
| H3 Ih (*532) | [5,3] | 120 | A5 × C2 | 10 | 15 |  |  |  |  |
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