Partial Derivation of Transformation Properties of Quarks and Leptons

Under the assumptions that $SU(3{)}_c\times U(1{)}_Y\times G^{\mathrm{\prime }}$ with $G^{\mathrm{\prime }}$ simple is a local symmetry group at high energies, that color is parity-conserving, and the Y-charges are irreducible, we show that anomaly constraints imply the minimal set of fermions is fifteen in number. Given this minimal set, we further show that $G^{\mathrm{\prime }}$ must be $SU(2)$ and the unbroken gauge symmetry is {\it either} color {\it or} the product of color with electric charge.