isomorphism of SL(2, Z2) and S3

Question

How can i prove that $SL_{2}(ℤ_{2})\cong S_3$?

It is easy to build explicit isomorphism but I think there is more beautiful solution.

Answer 1

To elaborate on what Albert said :

1. First calculate the order of $SL(2,\mathbb Z_2),$ like here.
2. Find two matrices in $SL(2,\mathbb Z_2)$ which do not commute.
3. Use the fact that the only nonabelian group of order 6 is $S_3,$ and this is given here.

Here the crucial step is (3) and this technique is specific to lower order groups.