Unitriangular matrix group:UT(3,Z4)

As a matrix group
This group is defined as the member of family::unitriangular matrix group of degree three over ring:Z4. Explicitly, it is the group (under matrix multiplication) of upper-triangular $$3 \times 3$$ unipotent matrices over the ring $$\mathbb{Z}/4\mathbb{Z}$$, i.e., matrices of the form:

$$\left\{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} : a_{12}, a_{13}, a_{23} \in \mathbb{Z}/4\mathbb{Z} \right\}$$

Definition by presentation
This group is given by the following presentation:

$$G := \langle x,y,z \mid x^4 = y^4 = z^4 = e, [x,y] = z, xz = zx, yz = zy\rangle$$

Here's how this presentation relates to the matrix description:

We can relate this with the matrix group definition by setting:

$$x = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}, \qquad y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\end{pmatrix}, \qquad z = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$$

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)));; gap> IdGroup(G); [ 64, 18 ]

Description by presentation
gap> F := FreeGroup(3);  gap> G := F/[F.1^4,F.2^4,F.3^4,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)];  gap> IdGroup(G); [ 64, 18 ]