Representations and classifications of compact gauge groups for unified weak and electromagnetic interactions

Assuming that all quarks have only electric charges $\frac{2}{3}$ and $-\frac{1}{3}$, we classify all possible representations and all admissible gauge groups $G$ underlying the unified weak and electromagnetic interactions. In particular, the group $G$ cannot contain any exceptional Lie groups ${\mathrm{G}}_2$, ${\mathrm{F}}_4$, ${\mathrm{E}}_6$, ${\mathrm{E}}_7$, and ${\mathrm{E}}_8$ as its factor. Moreover, the underlying irreducible representation for quark multiplets to be used must be one of fundamental representations for each component of simple Lie groups contained as a factor of $G$. For example, only the spinor representation is allowed for the $\mathrm{SO}\mathrm{}(2n+1)\mathrm{}$ group, while only the basic representation is admissible for the symplectic groups. If $G$ is semisimple in addition, then $G$ must be a product of $\mathrm{SU}\mathrm{}\left(3l\right)\mathrm{}$ groups.