  We perform a comprehensive analysis of a version of the 3-Higgs doublet model
whose scalar potential is invariant under a global $\Delta(27)$ discrete
symmetry and where the three scalar doublets are chosen to transform as a
triplet under this discrete group. For each of the known tree-level minima we
study the mass spectra and use the oblique parameters $STU$ as well as
perturbative unitarity to constrain the parameter space of the model. We then
discuss phenomenological consequences of some leading order flavour mixing
quark Yukawa couplings by considering the flavour violation process $b \to s
\gamma$.
  We show that perturbative unitarity significantly constrains parameters of
the model while, conversely, the beyond the Standard Model contributions to the
$b \to s \gamma$ decay are automatically tamed by the symmetry.
