Unitriangular matrix group of degree three over quotient of polynomial ring over F2 by square of indeterminate

As a matrix group
This group is defined as the unitriangular matrix group of degree three over the ring $$\mathbb{F}_2[t]/(t^2)$$. Explicitly, it is the group (under matrix multiplication) of upper-triangular $$3 \times 3$$ unipotent matrices over the ring $$\mathbb{F}_2[t]/(t^2)$$, i.e., matrices of the form:

$$\left\{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} : a_{12}, a_{13}, a_{23} \in \mathbb{F}_2[t]/(t^2)\right\}$$