Polyhedral group

Tetrahedral group Order 12 (24)	Octahedral group Order 24 (48)	Icosahedral group Order 60 (120)

T_d, [3,3], *332	O_h, [4,3], *432	I_h, [5,3], *532
T, [3,3]⁺, 332	O, [4,3]⁺, 432	I, [5,3]⁺, 532

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three polyhedral groups:

The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron.
The octahedral group of order 24, rotational symmetry group of the cube and the regular octahedron.
The icosahedral group of order 60, rotational symmetry group of the regular dodecahedron and the regular icosahedron.

These symmetries double to 24, 48, 120 respectly for the full reflectional groups.

See also

References

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973. (The Polyhedral Groups. §3.5, pp. 46–47)

External links

Weisstein, Eric W., "PolyhedralGroup" from MathWorld.