Gauged supergravities from Bianchi's group manifolds

We construct maximal D = 8 gauged supergravities by the reduction of D = I I supergravity over three-dimensional group manifolds. Such manifolds are classified into two classes, A and B, and eleven types. This Bianchi classification carries over to the gauged supergravities. The class A theories have 1/2 BPS domain wall solutions that uplift to purely gravitational solutions consisting of 7D Minkowski and a 4D Euclidean geometry. These geometries are generically singular. The two regular exceptions correspond to the near-horizon limit of the single- or double-centre Kaluza-Klein monopole. In contrast, the class B supergravities are defined by a set of equations of motion that cannot be integrated to an action and allow for no 1/2 BPS domain walls.