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The free fermionic construction of the heterotic string in four dimensionsproduced 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 asubgroup directly at the string scale. Over the past few years free fermionicmodels 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 $SU(4)\timesSU(2)_L\times U(1)_R$ (SU421) subgroup. The subspace of free fermionic modelsthat we consider corresponds to symmetric ${\mathbb{Z}}_2 \times{\mathbb{Z}}_2$ orbifolds. We provide a general argument that shows that thisclass of SU421 free fermionic models cannot produce viable three generationmodels.

Classification of SU(4) X SU(2) X U(1) Heterotic-String Models