Central product of UT(3,Z) and Z identifying center with 2Z

Definition
This group can be defined in the following equivalent ways:


 * 1) It is the central product of unitriangular matrix group:UT(3,Z) and the group of integers $$\mathbb{Z}$$ where the subgroup $$2\mathbb{Z}$$ of $$\mathbb{Z}$$ is identified with the center of $$UT(3,\mathbb{Z})$$.
 * 2) It is the following group of matrices under multiplication:

$$\left\{ \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\\end{pmatrix} \mid a,c \in \mathbb{Z}, 2b \in \mathbb{Z} \right \}$$

This group is almost like unitriangular matrix group:UT(3,Z). In fact, $$UT(3,\mathbb{Z})$$ occurs as a subgroup of index two inside it. However, unlike $$UT(3,\mathbb{Z})$$, it is a CS-Baer Lie group, and hence can participate in the CS-Baer correspondence.