Find the number of conjugacy classes of elements of order $2$ in $D_{2p}$ where $p$ is odd

Question

Find the number of conjugacy classes of elements of order $2$ in $D_{2p}$ where $p$ is odd

So I am trying to solve this problem, and I know that if $p$ is odd and

$$D_{2p}=\langle r, s \mid s^2=1, r^p=1 , srs^{-1}=r^{-1}\rangle $$

then the only element of order $2$ in $D_{2p}$ is $s$, so I guess that there is only one conjugacy class of these elements.

Am I wrong? If yes, could anyone explain it to me?

Answer 1

Or alternatively, if we want $D_{2p}$ to be $\langle r, s \mid s^2=1, r^{p}=1 , srs=r^{-1}\rangle $

So that, for example $D_{10}$ is the ten symmetries of the pentagon.

Then all the ways to flip a pentagon have order two, because you can just flip twice to get back to where you were.

And all those flips are along axes which go through one corner and the midpoint of the opposite side. They're all conjugate because you can rotate them into each other.

None of the rotations have order 2.

So just one class, and it's the same for all the other odd polygons.