Up to a rotation, how many ways can the faces of a cube be colored with three different colors?
Hint. The group of rigid motions of the cube can be described by the allowable permutations of the six faces and is isomorphic to S4. There are the identity cycle, 6 permutations with the structure (abcd) that correspond to the quarter turns, 3 permutations with the structure (ab)(cd) that correspond to the half turns, 6 permutations with the structure (ab)(cd)(ef) that correspond to rotating the cube about the centers of opposite edges, and 8 permutations with the structure (abc)(def) that correspond to rotating the cube about opposite vertices.