# Is there a “natural” transitive action of $SL_2(\mathbb{F}_5)$ on a set with 5 elements?

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.

GAP4, Version: 4.4.12 of 17-Dec-2008, x86_64-unknown-linux-gnu-gcc

• 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