We construct an ${\mathrm{SU}(3)}_{L}\ifmmode\times\else\texttimes\fi{}{U(1)}_{N}$ gauge model based on an ${S}_{2L}$ permutation symmetry for left-handed $\ensuremath{\mu}$ and $\ensuremath{\tau}$ families, which provides the almost maximal atmospheric neutrino mixing and the large solar neutrino m...