$S_{2L}$ permutation symmetry for left-handed $\mu$ and $\tau$ families and neutrino oscillations in an ${\mathrm{SU}\left(3\right)\mathrm{}}_L\times {U\left(1\right)\mathrm{}}_N$ gauge model

We construct an ${\mathrm{SU}\left(3\right)\mathrm{}}_L\times {U\left(1\right)\mathrm{}}_N$ gauge model based on an $S_{2L}$ permutation symmetry for left-handed $\mu$ and $\tau$ families, which provides the almost maximal atmospheric neutrino mixing and the large solar neutrino mixing of the large mixing angle type. Neutrinos acquire one-loop radiative masses induced by the radiative mechanism of the Zee type as well as tree level masses induced by the type II seesaw mechanism utilizing interactions of lepton triplets with an $\mathrm{SU}\left(3\right)\mathrm{}$-sextet scalar. The atmospheric neutrino mixing controlled by the tree-level and radiative masses turns out to be almost maximal owing to the presence of $S_{2L}$ supplemented by a $Z_4$ discrete symmetry. These symmetries ensure the near equality between the ${\nu }_e-{\nu }_{\mu }$ and ${\nu }_e-{\nu }_{\tau }$ radiative masses dominated by contributions from heavy leptons contained in the third members of lepton triplets, whose Yukawa interactions conserve $S_{2L}$ even after the spontaneous breaking. The solar neutrino mixing controlled by radiative masses, including a ${\nu }_{\mu }-{\nu }_{\tau }$ mass, which are taken to be of similar order, turns out to be described by large solar neutrino mixing angles.