Suppose I like combinatorics, and want to count how many ways to paint the faces of a tetrahedron using a pallet of $x$ colors.
I don't want to over count cases where I could just rotate one painted tetrahedron to look like another. So my idea is that you could find all the cycle decompositions of the elements in the group of symmetries of the tetrahedron, paint each cycle any of the $x$ colors, add them all up, and divide by the number of elements in the group, to count the distinct ways to paint, correct?
So my question boils down as, what is the group of symmetries of the tetrahedron? I would be happy just to know the elements in terms of rotations and flips around axes, and could probably figure out the cycle decompositions from there.