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 octahedral group is implemented in the Wolfram Language as FiniteGroupData["Octahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Oh", "PermutationGroupRepresentation"].
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 pure rotational octahedral subgroup is implemented in the Wolfram Language as a point group as FiniteGroupData["CrystallographicPointGroup", "O", "PermutationGroupRepresentation"].
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 .