Flipped SU(6) Unification of the $SU(3)_c\times SU(3)_L\times U(1)_X$ Model

We propose to partially unify the sequential $SU(3)_c\times SU(3)_L\times U(1)_X$ model (with $\beta=1/\sqrt{3}$) into the flipped $SU(6)$ model with the gauge group $SU(6)\times U(1)_K$. Gauge anomaly cancellation can easily be satisfied. We discuss the relevant Higgs sector, the low energy $331$ model spectrum and the unification of $SU(3)_c$ and $SU(3)_L$ gauge couplings. Neutrino masses generation and successful gauge coupling unification can set lower/upper bounds on the $331$ breaking scale. The partial proton decay lifetime of various channels, for example, the $p\to e^+ \pi^0$ channel, in flipped SU(6) GUT are discussed. We find that certain parameter region with $M_{331}\sim 10^{15}$ GeV of case II (for case with $M_{331}$ scale ${\bf \tilde{H}_{3,8}}$ Higgs field) can predict a partial proton lifetime of order $10^{34}$ years for $p\to e^+ \pi^0$ mode, which can be tested soon by future DUNE and Hyper-Kamiokande experiments.