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Determine the smallest symmetric group for this condition

I'm having trouble finding a subgroup of $SL_2(F_5)$ isomorphic to $H$, with $H$ generated by $x^4=y^3=1$, $xy=y^2x$.

My first thought went to upper triangular matrices, but that seems like going in the dark... Any hints?

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    Do you mean $H$ is the group $\langle x,y |x^4=y^3=1, xy=y^2x\rangle$?2012-11-13
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    @ChrisEagle Yes I do.2012-11-13
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    @ChrisEagle Could you give an example of such an element?2012-11-13
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    Not offhand, no.2012-11-13
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    @ChrisEagle Ok, well thanks anyway.2012-11-13
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    @Chris Eagle: $H$ is not the dihedral group of order 12. Like ${\rm SL}_2(5)$, $H$ has a unique element of order 2, whereas dihedral groups have lots of them.2012-11-13
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    Oops, yes, sorry.2012-11-13
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    You just asked this question yesterday, [Determine the smallest symmetric group for this condition](http://math.stackexchange.com/questions/236138/determine-the-smallest-symmetric-group-for-this-condition)2012-11-14

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Elements of order 3 have minimal polynomial $x^2+x+1$, so they have trace $-1$. In fact ${\rm SL}_2(5)$ has a unique conjugacy class of such elements, and so you can choose any such element for $y$, say $y=\left(\begin{array}{rr}0&1\\-1&-1\\ \end{array}\right)$.

Elements of order 4 have minimal polynomial $x^2+1$ an hence trace $0$. So you could let $x = \left(\begin{array}{rr}a&b\\c&-a\\ \end{array}\right)$,

where $a^2+bc=-1$, and then plug $x$ and $y$ into the equation $xy=y^2x$ and solve for $a,b,c$.