Subgroups of Symmetric groups isomorphic to dihedral group
Let $\left\{a_1,a_2,\ldots,a_n\right\}$ denote the vertices of an $n$-gon. The cyclic permutation $\alpha=(a_1,a_2,\ldots,a_n) \in S_n$. Choose any vertex, e.g. $a_1$ and consider it the fixed point of a reflexion of the plane (i.e the permutation $\beta=(a_2,a_n)(a_3,a_{n-1})\ldots$ of order $2$). This permutation also $\in S_n$. Now $D_n$ is generated by $\alpha, \beta \in S_n$ which shows that the $D_n$ thus defined is a subgroup of $S_n$