Show that there are two ways to embed a regular tetrahedron in a cube $C$ so that the vertices of the tetrahedron are also vertices of $C$. Show that the symmetry group of $C$ permutes these tetrahedra and deduce that the symmetry group of $C$ is isomorphic to the Cartesian product $S_{4} \times C_{2}$ of the symmetric group $S_{4}$ and the cyclic group $C_{2}$.