By the Pati-Salam group I refer to SU(2) x SU(2) x SU(4). It can be obtained as the group of isometries of the 8 dimensional manifold $S^3 \times S^5$, but I wonder if this is the only 8 dimensional manifold having this group of isometries.
This particular manifold is interesting because a quotient by any U(1) will produce a 7 dimensional manifold whose isometry group is the unbroken standard model group, as pointed out by Witten time ago. But my particular curiosity comes because Non Commutative Geometry gets the Pati Salam group from a different setup: the finite algebra
$M_2(H) \oplus M_4(C)$, related perhaps to deformations of even spheres.