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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: – t.b. Mar 29 '12 at 17:28

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)));

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 $Q_8$. – Joe Mar 29 '12 at 18:44

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