Dihedral Group

The symmetry group of a regular $n$-gon with corners labelled $1,2,\dots ,n$ has $2n$ elements and is written $D_{2n}=\{e,\rho ,{\rho }^2,\dots ,{\rho }^{n-1},{\sigma }_1,\dots ,{\sigma }_n\}$, where $$\begin{gathered} \rho =(1 \\ ,\dots n) \end{gathered}$$ and the ${\sigma }_i$ are reflections. This is the dihedral group of order $2n$, which is a subgroup of $S_n$.