  We formulate a predictive model of fermion masses and mixings based on a
$\Delta(27)$ family symmetry. In the quark sector the model leads to the viable
mixing inspired texture where the Cabibbo angle comes from the down quark
sector and the other angles come from both up and down quark sectors. In the
lepton sector the model generates a predictive structure for charged leptons
and, after radiative seesaw, an effective neutrino mass matrix with only one
real and one complex parameter. We carry out a detailed analysis of the
predictions in the lepton sector, where the model is only viable for inverted
neutrino mass hierarchy, predicting a strict correlation between $\theta_{23}$
and $\theta_{13}$. We show a benchmark point that leads to the best-fit values
of $\theta_{12}$, $\theta_{13}$, predicting a specific $\sin^2\theta_{23}
\simeq 0.51$ (within the $3 \sigma$ range), a leptonic CP-violating Dirac phase
$\delta \simeq 281.6 ^\circ$ and for neutrinoless double-beta decay $m_{ee}
\simeq 41.3$ meV. We turn then to an analysis of the dark matter candidates in
the model, which are stabilized by an unbroken $\mathbb{Z}_2$ symmetry. We
discuss the possibility of scalar dark matter, which can generate the observed
abundance through the Higgs portal by the standard WIMP mechanism. An
interesting possibility arises if the lightest heavy Majorana neutrino is the
lightest $\mathbb{Z}_2$-odd particle. The model can produce a viable fermionic
dark matter candidate, but only as a feebly interacting massive particle
(FIMP), with the smallness of the coupling to the visible sector protected by a
symmetry and directly related to the smallness of the light neutrino masses.
