So I have this question to solve. I've already shown that the group of rotations of a cube is isomorphic to $S_4$. I need to prove that these two groups are not conjugate when considered as subgroups of the group of isometries of 3D space. First off, I'm having a hard time figuring out how $S_4$ can be a subgroup of the group of isometries of 3D space. All the isometries in 3D can be represented as a product of reflections, so, this group would be a group containing the reflections in 3D. Obviously, I can picture this as $S_4$ representing the symmetries of a cube, but I don't see how that would help me solve the question. Could anybody give me some tips as to how to approach this question?
Presumably the question is referring to the other faithful representation of $S_4$ in three dimensions, namely the group of all symmetries of a regular tetrahedron.