The cube has 8 vertices, 12 edges and 6 faces, each of which is a square. There are rotational symmetries of order 2 (about the green axis), order 3 (the red axis) and order 4 (the blue axis).
The cube is also invariant under multiplication by -1, so
Dir(Cube) = Symm(Cube) x {1,-1}.
There are 4 long diagonals (shown below), which are permuted by the
action of the symmetry group, giving rise to a homomorphism
f:Symm(Cube) -> S4. The rotations
of orders 2, 3 and 4 are sent to 2-cycles, 3-cycles and 4-cycles
respectively. It turns out that
f is an isomorphism.
If we start with a cube (such as the wire frame below) and find the centres of all the faces we get the vertices of an octahedron. In other words, the dual of a cube is an octahedron.