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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): http://math.stackexchange.com/questions/233939/dicyclic-group-as-subgroup-of-s-6/233994#2339942012-11-13
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    @Micah Could you elaborate? Do you mean there is no such $S_n$?2012-11-13
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    $H$ 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 $S_n$.2012-11-13
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    @Micah is it $S_{12}$?2012-11-13
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    No it's not $S_{12}$. You know from an earlier question that $n>6$, so why not try $n=7$.2012-11-13
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    @DerekHolt y must have cycle type $(abc)(def)....$ so on with disjoint 3-cycles right? So $y^2$ must be $(acb)(dfe)....$ and then the conjugation necessary works out to be $(bc)(ef)...$ I'm not sure where I'm getting it wrong. –2012-11-13
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    Try $y$ with a single cycle $(a,b,c)$.2012-11-13
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    @DerekHolt Oh righttt why didn't I think of that. Thank you! Any hint on the subgroup of $SL_2(F_5)$?2012-11-13

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