A note on non-diagonalizable vertex-primitive digraphs

In the 1980s, P. J. Cameron asked whether the adjacency matrix of any arc-transitive digraph is diagonalizable. Babai gave a negative answer to this question and posed a problem regarding the existence of vertexprimitive digraphs whose adjacency matrices are non-diagonalizable. Recently, Li, Xia, Zhou and Zhu constructed an infinite family of such digraphs and gave some finite examples which are not Cayley digraphs. Among these finite examples was a digraph arising from the action of PSL2(17) on 2-subsets of the projective line PG(1)(17). In this note, we show that the smallest vertex-primitive digraphs whose adjacency matrices are non-diagonalizable arise from the action of PSL2(17) on 2subsets of PG(1)(17). We also show that the automorphism group of a non-diagonalizable vertex-primitive digraph is of almost simple type, diagonal type, or product type.