The Octahedron

The octahedron has 6 vertices, 12 edges and 8 faces, each of which is an equilateral triangle. The dual of an octahedron is a cube:

As dual polyhedra have the same symmetry groups, we have

         Symm(Oct) = Symm(Cube) = S₄.

Here we will just exhibit the three types of rotation axis for an octahedron.