  We consider an extension of the standard model with three Higgs doublet model
and $S_3\times \mathbb{Z}_2$ discrete symmetries. Two of the scalar doublets
are inert due to the $\mathbb{Z}_2$ symmetry. We have calculated all the mass
spectra in the scalar and lepton sectors and accommodated the leptonic mixing
matrix as well. We also show that the model has scalar and pseudoscalar
candidates to dark matter. Constraints on the parameters of the model coming
from the decay $\mu\to e\gamma$ were considered and we found signals between
the current and the upcoming experimental limits, and from that decay we can
predict the one-loop $\mu\to ee\bar{e}$ channel.
