Modular flavor symmetries of three-generation modes on magnetized toroidal orbifolds

We study the modular symmetry on magnetized toroidal orbifolds with Scherk-Schwarz phases. In particular, we investigate finite modular flavor groups for three-generation modes on magnetized orbifolds. The three-generation modes can be the three-dimensional irreducible representations of covering groups and central extended groups of ${\mathrm{\Gamma }}_N$ for $N=3,4,5,7,8,16$, that is, covering groups of $\mathrm{\Delta }(6(N\mathrm{/}2{)}^2)$ for $N=$ even and central extensions of $PSL(2,{\mathbb{Z}}_N)$ for $N=$odd with Scherk-Schwarz phases. We also study anomaly behaviors.