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
Best Answer
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.