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, [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, , 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, Th, 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, , 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, , include also each with inversion: • inversion • 12 × rotoreflection by 108°, order 10 • 12 × rotoreflection by 36°, order 10 • 20 × rotoreflection by 60°, order 6 • 15 × reflection, order 2 == Chiral polyhedral groups ==