TITLE: Rotation of group of rigid motions of cube is isomorphic to $S_{4}$
QUESTION [7 upvotes]: How can we show that a rotation of the group of rigid motions of the cube is isomorphic to $S_{4}$? This is what I have done:
The group of rigid motions of the cube can be assorted to 4 cases.
First, rotation by the diagonal.
Second, rotation by the line connecting the two centers of the opposite line.
Third, rotation by the line connecting the two centers of the opposite face.
Fourth, identity.

REPLY [3 votes]: Hints:
== Label each pair of opposite vertices in the cube as $\,1,2,3,4\,$ 
== Verify that each rigid motion of the cube maps each such pair of oppositve vertices to itself and, thus, you can recognize already an action of $\,S_4\,$ on the set of oppositive vertices.
== Verify the above action is transitive and faithful (injective), meaning: only the trivial motion fixes each pair.