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 
.
