5
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I'm really looking for a "cute" way of showing that $SL_2(\mathbb{F}_5)$ is a double cover of $A_5$. The sort of action I am looking for is something like the action of $GL_2(\mathbb{F}_3)$ on $\mathbb{P}^1(\mathbb{F}_3)$, which shows $GL_2(\mathbb{F}_3)$ is a double cover of $S_4$. Now that's cute.

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    Here's a related thread you may be interested in: http://math.stackexchange.com/q/93762/2012-03-29

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You can ask this type of questions to GAP:

GAP4, Version: 4.4.12 of 17-Dec-2008, x86_64-unknown-linux-gnu-gcc gap> g := SL(2,5);; gap> 5 in List(ConjugacyClassesSubgroups(g), c -> Index(g, Representative(c))); true gap>  

There is in fact a unique conjugacy class of subgroups of index 5, isomorphic to SL(2,3).

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    Smart! I was looking at conjugacy classes of elements but didn't think of looking at the subgroup lattice. There's also a conjugacy class of subgroups isomorphic to the quaternion group $Q_8$.2012-03-29