Show that $G=S_3S_5\rtimes S_2$ where $|G|=30$ and $S_p$ is a $p$-Sylow subgroup

Question

Let $G$ be a group of order $30$ and let $S_p<G$ a $p$-Sylow subgroup for $p=2,3,5$. Show that $G=S_3S_5\rtimes S_2$ (interior semi direct product).

I managed to show that $S_3S_5\cap S_2=\{e\}$ and that $S_3S_5\triangleleft G$ but I'm not being able to show that $G=S_3S_5S_2$.

Things I've proven that might be useful:

1. $|S_3S_5|=15$ thus $S_3S_5$ is a cyclic group.
2. $S_3\triangleleft G\lor S_5\triangleleft G$.
3. $S_2, S_3$ and $S_5$ are all cyclic (of course)

I tried assuming $g=a^mb^nc^r$ where $g$ is an arbitrary element of $G$ and $\langle a\rangle =S_2$, $\langle b\rangle=S_3$, $\langle c\rangle=S_5$ and $m,n,r\in\mathbb{Z}$ but didn't seem to lead anywhere.