Geometric interpretation of conjugacy classes and class equation of $D_6$

Question

Known that for Dihedral Group $D_6$, where $D_6=\{r,s: r^6=s^2=1, rs=sr^{-1}\}$, its conjugacy classes are given by $\{1\}, \{r,r^5\}, \{r^2,r^4\}, \{r^3\}, \{s, sr^2, sr^4\}, \{sr, sr^3, sr^5\}$, and its class equation is $12=1+1+2+2+3+3$, how can I interpret this result geometrically in terms of the symmetries of a regular hexagon?

Answer 1

There are two kinds of flip (or, reflection); the axis of a flip can be the line joining two opposite vertices, or it can be the line that's a perpendicular bisector of two opposite sides. The three flips of the first type form one conjugacy class, the three of the second type form another.

The two rotations one-sixth of the way around (one clockwise, one counterclockwise) form a conjugacy class; the two rotations one-third of the way around form another; the rotation halfway around is in a class by itself; and finally there's the identity.