# Determine the smallest symmetric group for this condition

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.

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My answer to this question can also be read as a hint for a): math.stackexchange.com/questions/233939/… – Micah Nov 13 '12 at 2:27
@Micah Could you elaborate? Do you mean there is no such $S_n$? – Benjamin Lu Nov 13 '12 at 4:25
$H$ is a finite group, so it certainly embeds in $S_n$ for some $n$ (by Cayley's theorem). I think if you experiment with some simple cycle types for $y$, and how it's possible for $x$ to conjugate $y$ appropriately while still having order $4$, it shouldn't be too tough to come up with the smallest possible $S_n$. – Micah Nov 13 '12 at 4:29
@Micah is it $S_{12}$? – Benjamin Lu Nov 13 '12 at 8:24
No it's not $S_{12}$. You know from an earlier question that $n>6$, so why not try $n=7$. – Derek Holt Nov 13 '12 at 8:35