Integer Heisenberg group
= Integer Heisenberg group
{c}
{title2=$H(\mathbb Z)$}
The integer Heisenberg group consists of upper-unitriangular three-by-three integer matrices. In coordinates $(x,y,z)$ its multiplication is
$$
(x,y,z)(x',y',z')=(x+x',y+y',z+z'+xy').
$$
It is the semidirect product $\mathbb Z^2\rtimes_A\mathbb Z$ for
$$
A=\begin{pmatrix}1&0\\1&1\end{pmatrix},
$$
and its <abelianization> is $\mathbb Z^2$.