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.

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.

The conjugacy classes of O are:

identity

6 × rotation by 90°, order 4

8 × rotation by 120°, order 3

3 × rotation by 180°, order 4

6 × rotation by 180°, order 2

The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron.

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}, 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}, 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} include also each with inversion: