We consider an extension of the standard model with three Higgs doublet model and $S_3times mathbb{Z}_2$ 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 $muto egamma$ were considered and we found signals between the current and the upcoming experimental limits, and from that decay we can predict the one-loop $muto eebar{e}$ channel.
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