  Leptogenesis appears to be a viable alternative to account for the baryon
asymmetry of the universe through baryogenesis. In this context, we consider a
scenario in which the standard model is extended with $S_3$ and $Z_2$ symmetry
in addition to the two scalar triplets, two scalar doublets and three right
handed neutrinos. Presence of scalar triplets and right-handed neutrinos in the
scenarios of both type-I and type-II seesaw framework provide a different
leptogenesis option and can help us to understand the matter-antimatter
asymmetry with simple $S_3$ symmetry. We discuss the neutrino phenomenology and
leptogenesis in both high ($O(10^{10})$ GeV) and low energy scale ($O$(2)TeV)
by constraining the Yukawa couplings. Moreover, we also consider the
constraints on model parameters from neutrino oscillation data and leptogenesis
to explain the rare lepton flavor violating decay and muon g-2 anomaly.
