Rotational symmetries of cube
There are 48 symmetries of the cube: 24 rotational x 2 reflectional.
The twenty three non-identity rotational symmetries are decomposed in the following graphic. The animated images are examples of particular axis of rotation.
Non-identity rotational symmetries of cube
Post script:
See Yaroslav’s comment below. The representation of these symmetries using CayleyGraph[SymmetricGroup[4]] is shown below.
Thank you Yaroslav.
Categories: Mathematica, Mathematics
Unknown Blogger Mathematica 
That’s a cool way of visualizing 2-cycle, 3-cycle and 4-cycle generators of the symmetric group. Only two generators are needed, so you can get any rotation of the cube by picking two axes from the 3 above, and combining several rotations of those two axes. Standard choice is 2-cycle,3-cycle, so if you do CayleyGraph[SymmetricGroup[4]], it’ll show you the map where red arrow represents 180 degree rotation (2-cycle), blue arrow gives 120 degree (3-cycle)
Thank you Yaroslav for this insight. I thought Mathematica provided a nice way of visualizing invariance under trasnformation and coudl show the axes of the 2,3,and 4-cycles.