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Wikipedia / 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, [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

Full polyhedral groups

See also

References