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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.