For connected nilpotent groups, 7 is the lowest dimension
where there are infinitely many non-isomorphic groups, and also
where some groups have no discrete cocompact subgroups. Here
one infinite family of 7-dimensional connected groups is studied,
discrete cocompact subgroups H are found for some of them, and then
the faithful simple quotients A of $C^{*}(\roman H)$ are identified. Such A are
shown to be isomorphic to $C^{*}$-crossed products $C^{*}(\roman H_
3,\Cal C(\Bbb T^3))$ generated
by some intriguing effective minimal distal flows $(\roman H_3,\Bbb T^
3)$, where $\roman H_3$ is the
discrete 3-dimensional Heisenberg group.