Let $G = \langle x,y,q ~|~ xy=qyx,~ qx = xq,~ qy=yq\rangle$ is an abstract group. 


1) Does this group have a common name? ("quantum tori" is the name of a certain related algebra)


2) What properties it has?


3) In particular, what subgroups it has?


Thanks.



Properties of quantum torus group
This is the integral Heisenberg group, the group of upper unitriangular integer $3\times 3$-matrices. It is nilpotent of class 2, and has many other properties. All subgroups can be described completely. 


