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].
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: