A congruence test for subgroups of $SL_2(Z)$

The subgroups of $SL_2(Z)$ can be divided in two types. There are the congruence subgroup, i.e. subgroups that contain a principal congruence subgroup $\Gamma(N)$, and the non-congruence subgroups. Often, the form a subgroup is given in does not allow an easy and direct determination of the type. In this talk I will present a method to check for a subgroup given by permutations if it is a congruence subgroup or not.