Integer Heisenberg group ($H(\mathbb{Z})$)

The integer Heisenberg group consists of upper-unitriangular three-by-three integer matrices. In coordinates $(x,y,z)$ its multiplication isIt is the semidirect product ${\mathbb{Z}}^2{⋊}_A\mathbb{Z}$ forand its abelianization is ${\mathbb{Z}}^2$.