Classification of symmetric toroidal orbifolds

We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield $\textitN$ $\geq$ 1 supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as $\mathbb{Z}_3$,$\mathbb{Z}_4$,$\mathbb{Z}_6$-I etc. and 358 with non-Abelian point groups such as S$_3$, D$_4$, A$_4$ etc. We also briefly explore the properties of some orbifolds with Abelian point groups and $\textit{N}$=1, i.e. specify the Hodge numbers and comment on the possible mechanisms (local or non-local) of gauge symmetry breaking