Conjugacy classes of rotational symmetry group of cube

Question

I have a cube with rotations $\{r, r^2, s, t\}$. I want to find the cardinality of the conjugacy classes for these elements. (I know they are 6, 3, 8 and 6 respectively) I couldn't find any formula or anything so I tried to do it by hand for $r$, which seemed to work (see left side of my note picture), but for $r^2$ I ended up with the same elements in its conjugacy class as in $r$. I haven't tried for $s$ or $t$. Or is there some algebra like there is for dihedral groups (like $sr^b=r^{-b}s$)

I also didn't know in which order I had to apply the elements ($s^2tr^2$ versus $r^2s^2t$ for example).

I am using this to calculate the distinct orbits of a group (correct terminology?) using the Counting Theorem

Answer 1

Wikipedia has a good explanation. This is basically the geometric explanation of what your elements are:

$r$: rotation about an axis from the center of a face to the center of the opposite face by an angle of 90°: 3 axes, 2 per axis, together 6

$r^2$: ditto by an angle of 180°: 3 axes, 1 per axis, together 3

$t$: rotation about an axis from the center of an edge to the center of the opposite edge by an angle of 180°: 6 axes, 1 per axis, together 6

$s$: rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8

and of course, don't forget the identity.

Answer 2

Hint: Maybe instead of thinking about the cube you can think about the diagonals. Can you recover the same information from the diagonals? If so, which group does this look like? With diagonals I mean line segments between pairs of points with distance sqrt(3) assuming the cube has edge length 1.
On a note, I believe r shouldn't be conjugate to r^2. You might want to check that.