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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.

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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.

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