Smith’s Theorem and a characterization of the 6-cube as distance-transitive graph

A generic distance-regular graph is primitive of diameter at least two and valency at least three. We give a version of Derek Smith's famous theorem for reducing the classification of distance-regular graphs to that of primitive graphs. There are twelve cases—the generic case, four canonical imprimitive cases that reduce to the generic case, and seven exceptional cases. All distance-transitive graphs were previously known in six of the seven exceptional cases. We prove that the 6-cube is the only distance-transitive graph coming under the remaining exceptional case.