  In this work we study a predictive model based on a partially unified theory
possessing the gauge symmetry of the Pati-Salam group, $SU(2)_L\times
SU(2)_R\times SU(4)_C$ supplemented by a global Peccei-Quinn symmetry,
$U(1)_{PQ}$. A comprehensive analysis of the Higgs potential is carried out in
a minimal set-up. The assumed Peccei-Quinn symmetry along with solving the
strong CP problem, can provide axion as the dark matter candidate. This minimal
set-up with limited number of Yukawa parameters can successfully incorporate
the hierarchies in the charged fermion masses and mixings. The automatic
existence of the heavy Majorana neutrinos generate the extremely small light
neutrino masses through the seesaw mechanism, which is also responsible for
producing the observed cosmological matter-antimatter asymmetry of the
universe. We find interesting correlation between the low scale neutrino
observables and the baryon asymmetry in this model. Baryon number violating
nucleon decay processes mediated by the scalar diquarks and leptoquarks in this
framework are found to be, $n,p\to \ell+m, \ell^c + m$ ($m=$ meson, $\ell=$
lepton, $\ell^c=$ antilepton) and $n,p\to \ell+\ell^c + \ell^c$. For some
choice of the parameters of the theory, these decay rates can be within the
observable range. Another baryon number violating process, the
neutron-antineutron oscillation can also be in the observable range.
