  We propose a 3+1 Higgs Doublet Model based on the $\Delta(27)$ family
symmetry supplemented by several auxiliary cyclic symmetries leading to viable
Yukawa textures for the Standard Model fermions, consistent with the observed
pattern of fermion masses and mixings. The charged fermion mass hierarchy and
the quark mixing pattern is generated by the spontaneous breaking of the
discrete symmetries due to flavons that act as Froggatt-Nielsen fields. The
tiny neutrino masses arise from a radiative seesaw mechanism at one loop level,
thanks to a preserved $Z_2^{\left( 1\right)}$ discrete symmetry, which also
leads to stable scalar and fermionic dark matter candidates. The leptonic
sector features the predictive cobimaximal mixing pattern, consistent with the
experimental data from neutrino oscillations. For the scenario of normal
neutrino mass hierarchy, the model predicts an effective Majorana neutrino mass
parameter in the range $3$~meV$\lesssim m_{\beta\beta}\lesssim$ $18$ meV, which
is within the declared range of sensitivity of modern experiments. The model
predicts Flavour Changing Neutral Currents which constrain the model, for
instance Kaon mixing and $\mu \to e$ nuclear conversion processes, the latter
which are found to be within the reach of the forthcoming experiments.
