Polyhedral group


In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

Groups

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, 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, 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:

Full polyhedral groups