Group of rigid rotations of cube

[PhDorBust]

I'm having trouble visualizing some of the rotations that compose this group.

Clearly the group has 24 elements by argument any of 6 faces can be up, and then cube can assume 4 different positions for each upwards face.

My book describes the rotations as follows:
3 subgroups of order 4 created by rotation about line passing through center of two faces.
4 subgroups of order 3 created by "taking hold of a pair of diagonally opposite vertices and rotating through the three possible positions, corresponding to the three edges emanating from each vertex."

My trouble lies with the second description, that is, I haven't the slightest idea of what it is saying. Any help? Also, any general comments on visualizing symmetry groups would be appreciated, I trouble going beyond dihedral group of order 4.

 

[tiny-tim]

Hi PhDorBust!

4 subgroups of order 3 created by "taking hold of a pair of diagonally opposite vertices and rotating through the three possible positions, corresponding to the three edges emanating from each vertex."

Any vertex has 3 edges …

rotate the cube about that vertex so as to interchange those 3 edges.

(there are 8 vertices altogether, but only 4 subgroups, because rotations about opposite vertices give the same effect)