Answer by vdelecroix for Hi,Beginner in Sage (I love it!) , I want to ask you this maybe naive question:I have a group $G$ defined by $\lbrace M \in SL_2(Z) \ | \ M = I + kB \ mod(N) \rbrace $, where $I$ is the identity, $N$ a given positive integer, $B$ is a given fixed matrix verifying $B^2 = 0$ and $k$ (not fixed) any integer in { 0,1,2,....,N-1 }. I want to explore conjugacy classes/subgroups of this group $G$ for some specific $B$ and $N$. Is this subgroup finitely generated for some $N$? subgroups of finite index? etc... Do I have a way to do that with SageMath?Remark thatIt is subgroup of a finitely generated group butv that do not imply that it is finitely generated$B^2 = 0$ gives that $I + kB = \Lambda^k \ with \ \Lambda = I + B$. An obvious subgroup is $ H = \lbrace \Lambda^k, \ k \in \mathbb Z \rbrace$Any advise or web pointer would be appreciatedThanks for your help

Here is a function constructing the group def 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 as sage: 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 at[SageMath documentation on subgroups of SL(2,Z)](http://doc.sagemath.org/html/en/reference/arithgroup/index.html) 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 works **EDIT:** 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 $\begin{pmatrix}0&b\\\\0&0\end{pmatrix}$.