  We study of fermion zero-modes on magnetized $T^6/\mathbb{Z}_N$ orbifolds. In
particular, we focus on non-factorizable orbifolds, i.e. $T^6/\mathbb{Z}_7$ and
$T^6/\mathbb{Z}_{12}$ corresponding to $SU(7)$ and $E_6$ Lie lattices
respectively. The number of degenerated zero-modes corresponds to the
generation number of low energy effective theory in four dimensional
space-time. We find that three-generation models preserving 4D $\mathcal{N}=1$
supersymmetry can be realized by magnetized $T^6/\mathbb{Z}_{12}$, but not by
$T^6/\mathbb{Z}_7$. We use $Sp(6,\mathbb{Z})$ modular transformation for the
analyses.

High Energy Physics - Theory;High Energy Physics - Phenomenology