There are five inscribed cubes in the dodecahedron. Each of these
cubes has twelve edges, with one edge cutting across each of the
twelve faces of the dodecahedron. If we focus on one of these
cubes, then the full dodecahedron can be obtained by attaching a
tent to each face of the cube. The rotational symmetry group of
the dodecahedron acts on the set of five cubes as the alternating
permutation group $A_5$.