  We investigate, for the first time, a scenario where the dark matter consists
of three complex scalar fields that are stabilized by a single $Z_7$ symmetry.
As an extension of the well-known scalar Higgs-portal, this $Z_7$ model is also
subject to important restrictions arising from the relic density constraint and
from direct detection experiments. Our goal in this paper is to find and
characterize the viable regions of this model, and to analyze its detection
prospects in future experiments. First, the processes that affect the relic
densities are identified (they include semiannihilations and conversions) and
then incorporated into the Boltzmann equations for the dark matter abundances,
which are numerically solved with micrOMEGAs. By means of random scans of the
parameter space, the regions consistent with current data, including the recent
direct detection limit from the LZ experiment, are selected. Our results reveal
that the $Z_7$ model is indeed viable over a wide range of dark matter masses
and that both conversions and semiannihilations play an important role in
determining the relic densities. Remarkably, we find that in many cases all
three of the dark matter particles give rise to observable signals in future
direct detection experiments, providing a suitable way to test this scenario.
