MSSM-like from $S{U}_5\times D_4$ Models

Using finite discrete group characters and symmetry breaking by hyperflux as well as constraints on top- quark family, we study minimal low energy effective theory following from SU${}_5\times D_4$ models embedded in F-theory with non abelian flux. Matter curves spectrum of the models is obtained from SU${}_5\times S_5$ theory with monodromy $S_5$ by performing two breakings; first from symmetric group $S_5$ to $S_4$ subsymmetry; and next to dihedral $D_4$ subgroup. As a consequence, and depending on the ways of decomposing triplets of $S_4$, we end with three types of $D_4$- models. Explicit constructions of these theories are given and a MSSM- like spectrum is derived.