Let $G$ be the subgroup of $S_5$ generated by the cycle $(12345)$ and the element $(15)(24)$. Prove that $G \cong D_5$, where $D_5$ is the dehidral group of order $10$.
I understand intuitively that $(12345)$ corresponds to an rotation in $D_5$ and $(15)(24)$ corresponds with a reflection in $D_5$. I would say let $\rho$ be a reflection in $D_5$ and $\sigma$ a reflection in $D_5$. Then $D_5$ is generated by $\sigma$ and $\rho$. Let $r=(12345), s=(15)(24)$.
Define $f: D_5 \to G : \rho ^i \circ \sigma ^j \mapsto r^i s^j$.
The thing where I get stuck is that I can't show that $r^i s^j$ has order $2$ if $j=1$. With reflections this is quite simple, as a any reflection has order $2$, but how do I show this with permutations ? Can anybody help me how I can construct a solid proof here ?