  We classify all complete projective special real manifolds with reducible
cubic potential, obtaining four series. For two of the series the manifolds are
homogeneous, for the two others the respective automorphism group acts with
co-homogeneity one. Complete projective special real manifolds give rise to
complete quaternionic K\"ahler manifolds via the supergravity q-map, which is
the composition of the supergravity c-map and r-map. We develop curvature
formulas for manifolds in the image of the q-map. Applying the q-map to one of
the above series of projective special real manifolds, we obtain a series of
complete quaternionic K\"ahler manifolds, which are shown to be inhomogeneous
(of co-homogeneity one) based on our curvature formulas.
