  In spite of the success of the Standard Model of Particle Physics, there are
some theoretical predictions which are not yet fully established experimentally
as well as some experimental observations which cannot be fitted within its
theoretical framework, thus requiring physics beyond the Standard Model. One of
these is a hypothetical non-luminous form of matter - dark matter. Models with
an extended scalar electroweak sector yield plausible dark matter candidates.
In this paper we study a specific model, C-III-a, from a family of
$S_3$-symmetric three-Higgs-doublet models. The model consists of two active
SU(2) doublets and an inert one. The latter is inert due to a $\mathbb{Z}_2$
symmetry that survives the breaking of $S_3$, and would accommodate a dark
matter particle. We explore the model numerically, based on theoretical and
experimental constraints. After applying a number of successive checks over the
parameter space we found a viable dark matter mass region in the range
$[6.5;\,44.5]~\text{GeV}$. This region is drastically different from the
Higgs-like dark matter states that have been proposed: the well-known Inert
Doublet Model and models with three scalar doublets, with one or two inert
doublets. Furthermore, the C-III-a model allows for spontaneous CP violation.
