  We consider the magnetic Laplacian $\Delta_{\nu,\mu}$ on
$\mathbb{R}^{2n}=\mathbb{C}^n$ given by $$ \Delta_{\nu,\mu}=
4\sum\limits_{j=1}\limits^{n}\frac{\partial^2 }{\partial z_j \partial
\overline{z_j}} +2i\nu (E+ \overline{E} +n) +2\mu (E- \overline{E} )
-(\nu^2+\mu^2)|z|^2. $$ We show that $\Delta_{\nu,\mu}$ is connected to the
sub-Laplacian of a group of Heisenberg type given by $\mathbb{C}\times_\omega
\mathbb{C}^n$ realized as a central extension of the real Heisenberg group
$H_{2n+1}$. We also discuss invariance properties of $\Delta_{\nu,\mu}$ and
give some of their explicit spectral properties.
