a) Determine the smallest symmetric group $S_n$ that contains a subgroup isomorphic to H, generated by $x^4=y^3=1$, $xy=y^2x$.
b) Find a subgroup of $SL_2(F_5)$ that is isomorphic to that group.
My first step was to say that $n$ is at least 4 so that $S_n$ contains an element of order 4. But I tried $S_4$ and it didn't quite work. After that I couldn't make much progress.
Any help appreciated.