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We construct a $D_4$ flavor model based on SU(3)_C X SU}(3_L X U(1)_X gaugesymmetry responsible for fermion masses and mixings. The neutrinos get smallmasses from antisextets which are in a singlet and a doublet under $D_4$. Ifthe D_4 symmetry is violated as perturbation by a Higgs triplet under SU(3)_Land lying in {1}''' of D_4, the corresponding neutrino mass mixing matrix getsthe most general form. In this case, the model can fit the experimental data in2012 on neutrino masses and mixing. Our results show that the neutrino massesare naturally small and a little deviation from the tribimaximal neutrinomixing form can be realized. The quark masses and mixing matrix are alsodiscussed. In the model under consideration, the CKM matrix can be differentfrom the unit matrix. The scalar potential of the model is more simpler thanthose of the model based on $S_3$ and $S_4$. Assignation of VEVs to antisextetsleads to the mixing of the new gauge bosons and those in the Standard Model.The mixing in the charged gauge bosons as well as the neutral gauge boson isconsidered.

The D_4 flavor symmery in 3-3-1 model with neutral leptons