Find representative for each conjugacy class of $D_{10}$

Question

How do you find representative for each conjugacy class of $D_{10} = \langle r \rangle_5 \rtimes \langle s \rangle_2$?

I know $D_{10}$ has $4$ conjugacy classes which are:

$[Id]$, $[r]$, $[r^2]$, $[s]$.

Answer 1

Since you have already mentioned a representative for each conjugacy class in your question, I assume, that you want to know all representatives for each conjugacy class. Here they are: $$[Id] = \{Id\}$$ $$[r] = \{r, r^4\}$$ $$[r^2] = \{r^2, r^3\}$$ $$[s] = \{s, rs, r^2s, r^3s, r^4s\}$$

More general information (concerning the solution of similar problem for $D_n$ with arbitrary $n$) can be found under this link: https://groupprops.subwiki.org/wiki/Dihedral_group#Conjugacy_class_structure