Let \(n\geq 1\). The dihedral group of order \(2n\) is the group with presentation \[ D_n \coloneqq \gens{r, s \st r^n, s^2, srs^{-1}r } . \]
For \(n\geq 1\), \[ \gens{r, s \st r^n, s^2, srs^{-1}r } \cong \gens{r, s \st r^n, s^2, (rs)^2 }, \] by an isomorphism sending \(r\mapsto r\) and \(s\mapsto s\).
For \(n\geq 3\), \(D_n\) is the group of symmetries of a regular \(n\)-gon in the plane, with \(r\) the rotation by \(2\pi/n\) and \(s\) a reflection. Labeling the vertices \(1, \ldots, n\) in cyclic order identifies \(D_n\) with a subgroup of \(S_n\), in which \(r\) is an \(n\)-cycle.
Label the vertices of a square \(1,2,3,4\) in cyclic order. Then \(D_4 = \gens{(1,2,3,4), (1,3)} \leq S_4\), where \((1,2,3,4)\) is the rotation by \(2\pi/4\) and \((1,3)\) is the reflection across the diagonal through the vertices \(2\) and \(4\).
Label the vertices of a regular pentagon \(1,\ldots,5\) in cyclic order. Then \(D_5 = \gens{(1,2,3,4,5), (1,5)(2,4)} \leq S_5\), where \((1,2,3,4,5)\) is the rotation by \(2\pi/5\) and \((1,5)(2,4)\) is the reflection across the line through vertex \(3\) and the center.