Octahedral symmetry

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of 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 pairs of opposite sides of the octahedron.

Chiral and full (or achiral) octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system.

As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product ${\displaystyle S_{2}\wr S_{3}\simeq S_{2}^{3}\rtimes S_{3}}$,
and a natural way to identify its elements is as pairs ${\displaystyle (m,n)}$ with ${\displaystyle m\in [0,3!)}$ and ${\displaystyle n\in [0,2^{3})}$.
But as it is also the direct product ${\displaystyle S_{4}\times S_{2}}$, one can simply identify the elements of ${\displaystyle S_{4}}$ as ${\displaystyle a\in [0,4!)}$ and their inversions as ${\displaystyle a'}$.

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