Lie-type-like groups

A finite simple group of Lie type in defining characteristic p has exactly two p-blocks, the principal block and a block of defect zero consisting of the Steinberg character whose degree is the p-part of the order of the group. In this paper we characterize finite groups G which have exactly the principal p-block and a p  -block of defect zero consisting of an irreducible character of degree |G|p|G|p.