Classification of $SU(4)\times SU(2)\times U(1)$ heterotic-string models

The free fermionic construction of the heterotic string in four dimensions produced a large space of three-generation models with the underlying $SO(10)$ embedding of the Standard Model states. The $SO(10)$ symmetry is broken to a subgroup directly at the string scale. Over the past few years free fermionic models with the Pati-Salam and flipped $SU(5)$ subgroups have been classified. In this paper we extend this classification program to models in which the $SO(10)$ symmetry is broken at the string level to the $SU(4)\times SU(2{)}_L\times U(1{)}_R$ (SU421) subgroup. The subspace of free fermionic models that we consider corresponds to symmetric ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ orbifolds. We provide a general argument that shows that this class of SU421 free fermionic models cannot produce viable three-generation models.