Conjugacy classes of the rotations of the cube

Question

Could anyone help calculate the conjugacy classes in the rotation group of the cube (in terms of the type of axis of rotation and angle of rotation)? I know that they are related to $S_4$, and the axis of rotation are the main diagonals of the cube, but how do I describe it?

Answer 1

The group of rotational symmetries (also sometimes referred to as the orientation preserving symmetries) is indeed isomorphic to $S_4$, the trouble is figuring what kind of isomorphism there is from the cube to the symmetric group. The cube has eight faces and six vertices so its not immediately clear where a set of four objects will come from. However, if one consider the set of opposing corner of the cube, you can convince yourself that there are in fact, four of them. Thus the rotations of the cube are precisely the permutations of there four sets of opposite corners (or diagonals if you like). If you're having a hard time seeing this, there's really nothing better than picking up a cube (I like to use tissue boxes since they're a decent size and can be written on) and start turning it around.

Once you've convinced yourself that this really is an isomorphism, you've reduced the problem to finding the conjugacy classes of $S_4$ which is trivially easy (think about cycle type).

If you're struggling with realizing the isomorphism from the cube to $S_4$, you can also think about the group of rotations of an octahedron, which is the same as that of a cube, since they are dual, although this need justification if you want to use it.