is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated octahedron. It has conjugacy classes 1, , , , , , , , , and (Cotton 1990). Its multiplication table is illustrated above.

The great rhombicuboctahedron can be generated using the matrix representation of using the basis vector .

The octahedral group has a pure rotation subgroup denoted that is isomorphic to the tetrahedral group . is of order 24 and has conjugacy classes 1, , , , and (Cotton 1990, pp. 50 and 434). Its multiplication table is illustrated above.

The cycle graph of is illustrated above.

Platonic and Archimedean solids that can be generated by group are illustrated above, with the corresponding basis vector summarized in the following table, where and are the largest positive roots of the cubic polynomials and .

solid	basis vector
cube
cuboctahedron
octahedron
small rhombicuboctahedron
snub cube
truncated cube
truncated octahedron

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 47-49, 1990.

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

Lomont, J. S. "Octahedral Group." §3.10.D in Applications of Finite Groups. New York: Dover, p. 81, 1987.

CITE THIS AS:

Weisstein, Eric W. "Octahedral Group." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OctahedralGroup.html