Hypercharge generators in SU(7) grand unification models

We present a thorough analysis of U(1) generators in all of the nine possible symmetry-breaking patterns that reduce SU(7) down to ${\mathrm{SU}}_c\mathrm{}\left(3\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$. There is one allowed representation that contains three generations of leptons and at least two generations of quarks and satisfies anomaly freedom, reality under ${\mathrm{SU}}_c\mathrm{}\left(3\right)\mathrm{}$ and ${\mathrm{U}}_{\mathrm{em}}$(1), and complexity under ${\mathrm{SU}}_c\mathrm{}\left(3\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$. In addition, three other representations, two having only one and two of the three lepton-quark generations complex with respect to ${\mathrm{SU}}_c\mathrm{}\left(3\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$, and the other having fractionally charged color singlets in pairs embedded in the fundamental representation [7], are also analyzed for completeness.