Let $D_n$ be the dihedral group of a regular $n-$polygon. I'd like to write $D_n$ as a subgroup of $S_n$, the set of all permutations of $n$ objects and therefore show why the order of $D_n$ is $2n$.
Clearly, I need to only consider the permutations of the vertices that preserve distances. However, I'm not quite sure how to go about listing these permutations.
Any advice would be helpful!