  We systematically investigate the finite set of possible gauge groups and
matter content for N = 1 supergravity theories in six dimensions with no tensor
multiplets, focusing on nonabelian gauge groups which are a product of SU(N)
factors. We identify a number of models which obey all known low-energy
consistency conditions, but which have no known string theory realization. Many
of these models contain novel matter representations, suggesting possible new
string theory constructions. Many of the most exotic matter structures arise in
models which precisely saturate the gravitational anomaly bound on the number
of hypermultiplets. Such models have a rigid symmetry structure, in the sense
that there are no moduli which leave the full gauge group unbroken.
