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

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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.

Arithmetic functions

Function	Value	Similar groups	Explanation
nilpotency class	2
derived length	2
Frattini length	2
Hirsch length	3
polycyclic breadth	3

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	Yes
group of nilpotency class two	Yes
metacyclic group	No
polycyclic group	Yes
metabelian group	Yes
supersolvable group	Yes