  The Yukawa and scalar sectors of a general $S_3$-symmetric three-Higgs
doublet model (3HDM) are investigated. The Yukawa interactions are constructed
in an $S_3$-invariant way, while the scalar potential contains $S_3$
soft-breaking terms. Global fits to the quark/lepton masses and CKM/PMNS
matrices are performed. Excellent fits to all fermion mass and mixing
parameters are obtained. Both normal ordering and inverted ordering of neutrino
masses are found to be admissible within the framework, with a prediction for
the CP-violation phase, $\delta_{CP} \simeq 120^0$. The fit results in the
Yukawa sector are further investigated, together with the scalar sector,
imposing constraints from Higgs-mediated neutral meson mixing and neutron
electric dipole moment (EDM). We explore the lowest allowed mass of the heavy
Higgs bosons, consistent with these constraints, and find it to be about 17
TeV. The corresponding neutron EDM is around $1.7\times10^{-27}$ e-cm, which is
within reach of proposed experiments. It is found that the constraints from the
$K$-meson system dominate, while those from the $D$ meson system are marginal.
