Polyhedral group

In geometry, the polyhedral groups are 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 A₄.
* 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 S₄.
* 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 A₅.
* 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, can be seen as the union of 6 tetrahedral symmetry mirrors, and 3 mirrors of dihedral symmetry Dih₂, . Pyritohedral symmetry is another doubling of tetrahedral symmetry.
The conjugacy classes of full tetrahedral symmetry,, are:

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

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

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