Comment by oldani for Here is a function constructing the groupdef my_congruence_group(B, N): M = MatrixSpace(Zmod(N), 2) B = M(B) assert (B**2).is_zero() G = SL(2, Zmod(N)) H = G.subgroup([1 + k*B for k in range(N)]) C = libgap.RightCosets(G, H) l = libgap.Permutation(G([1,1,0,1]), C, libgap.OnRight) r = libgap.Permutation(G([1,0,1,1]), C, libgap.OnRight) s2 = libgap.Permutation(G([0,-1,1,0]), C, libgap.OnRight) s3 = libgap.Permutation(G([0,1,-1,1]), C, libgap.OnRight) return ArithmeticSubgroup_Permutation(L=l.sage(), R=r.sage(), S2=s2.sage(), S3=s3.sage()) You can just copy paste the code and use it assage: N = 6 sage: B = [2,1,2,4] sage: G = my_congruence_group(B, N) sage: G.gens() # some generators for the group ( [1 3] [ 3 -2] [ 11 -25] [0 1], [ 2 -1], [ 4 -9] ) sage: print(G.genus()) # the genus of the quotient 0 sage: print(G.nu2(), G.nu3(), G.ncusps()) # elliptic points and cusps on the quotient 0 0 4 sage: F = G.farey_symbol() sage: F.fundamental_domain() # plot a fundamental domain You should have a look atSageMath documentation on subgroups of SL(2,Z) documentation to learn how to use G. in particular the section "Arithmetic subgroups defined by permutations of cosets" which is how I build your group and "Farey Symbol for arithmetic subgroups of PSL2(Z)" which teach you how constructing the fundamental domain worksEDIT: Though I suspect that all your groups are just conjugate to $\Gamma_1(N)$ or $\Gamma_0(N)$ (or at least close enough). The only matrix $B$ with $B^2= 0$ are conjugate to $\begin{pmatrix}0&b\\0&0\end{pmatrix}$.

Wonderful, thank you so much for your time and your great answer. You taught me a lot in this last two hours. I will study that with great pleasure.