Octahedral symmetry

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Selected point groups in three dimensions

Involutional symmetry Cs, (*) [ ] =	Cyclic symmetry Cnv, (*nn) [n] =	Dihedral symmetry Dnh, (*n22) [n,2] =
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry Td, (*332) [3,3] =	Octahedral symmetry Oh, (*432) [4,3] =	Icosahedral symmetry Ih, (*532) [5,3] =

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron.

The group of orientation-preserving symmetries is S₄, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube.