Unitriangular matrix group:UT(3,Z)

As a reduced free group
Abstractly, this group is a free class two group on a generating set of size two. Hence, it is a reduced free group.

As a matrix group
This group, denoted $$UT(3,\mathbb{Z})$$ or $$U(3,\mathbb{Z})$$, is defined as the unitriangular matrix group of degree three over the ring of integers. Explicitly, it is the group, under multiplication:

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

The group is also sometimes called the integral Heisenberg group.

Definition by presentation
The group can be defined by means of the following presentation:

$$\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy \rangle$$

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

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

Structures
The group has the structure of an has structure of::arithmetic group.