Classification of three-generation models by orbifolding magnetized <i>T</i>2 × <i>T</i>2

We study orbifolding by the $\mathbb{Z}_2^{\rm (per)}$ permutaion of $T^2_1 \times T^2_2$ with magnetic fluxes and its twisted orbifolds. classify possible three generation models which lead to non-vanishing Yukawa couplings on magnetized orbifolds including permutation (t)}$ twist. also modular symmetry such orbifold models. As an illustrating model, we examine realization quark masses mixing angles.