Unitriangular matrix group:UT(3,Z9)

As matrices
This group is defined as the unitriangular matrix group of degree three over ring:Z9. Explicitly, it is the group (under matrix multiplication) of upper-triangular $$3 \times 3$$ unipotent matrices over the ring $$\mathbb{Z}/9\mathbb{Z}$$, i.e., matrices of the form:

$$\left\{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} : a_{12}, a_{13}, a_{23} \in \mathbb{Z}/9\mathbb{Z} \right\}$$

Description as a matrix group
This description uses the functions SL, ZmodnZ, IsZero, and IsOne.

gap> L := SL(3,ZmodnZ(4));; gap> G := Group(Filtered(L,x -> ForAll([x[1][1],x[2][2],x[3][3]],IsOne) and ForAll([x[2][1],x[3][1],x[3][2]],IsZero)));;

Description by presentation
gap> F := FreeGroup(3);  gap> G := F/[F.1^9,F.2^9,F.3^9,F.1*F.3*F.1^(-1)*F.3^(-1),F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1)*F.1^(-1)]; 