The principal congruence subgroup $\Gamma(n)$ of $SL(2,\mathbb Z)$ is a normal subgroup of $SL(2,\mathbb Z)$ with index $$[SL(2,\mathbb Z):\Gamma(n)]=n^3\prod_{p|n}(1-\tfrac{1}{p^2}),$$ where the product is over all primes dividing $n$. The quotient group is isomorphic to $SL(2,\mathbb Z)/\Gamma(n)\cong SL(2,\mathbb Z/n\mathbb Z)$. Is this group known to be isomorphic to a family of basic groups, such as $S_n$, $A_n$ or $D_n$?
For $n=2$ the quotient group must be non-abelian and of order $6$, and thus isomorphic to the symmetric group $S_3$. Is it known for $n\geq 3$?