  Extensions of the Standard Model have been attempted from the bottom up and
from the top down yet there remains a largely unexplored middle ground. In this
paper, using the Mathematica package LieART, we exhaustively enumerate
embeddings of the Standard Model within the class of theories with
bifundamental fermions in product gauge group SU(a)XSU(b)XSU(c) with no more
generators than E6, while achieving SM family unification rather than
replication. We incorporate simple phenomenological constrains and find 151
unique models, including 9 that have only vector-like particle content
beyond-the-Standard Model (BSM) particles, which we conjecture belong to 5
infinite families of such models. We describe the potentially most viable
models: namely, the 9 with strictly vector-like BSM content along with the 29
models we found with no more than 30 additional BSM chiral particles. These
include models with fractional electric charge color singlets, and hence
magnetic monopoles with multiple Dirac charge. This latter collection of models
predicts chiral particles with masses near the electroweak scale accessible to
current and future collider experiments.
