Dihedral group/Subgroup of the permutation group/Alternating/Exercise

We consider a regular ${}n$-gon and the corresponding group of all (proper and improper) symmetries, that is, the dihedral group ${}D_{n}$. Describe ${}D_{n}$ as a subgroup of the permutation group ${}S_{n}$. Which permutations generate the dihedral group? For which ${}n$ is the dihedral group a subgroup of the alternating group?

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