Representations of the double Burnside algebra and cohomology of the extraspecial p-group

Let E be the extraspecial p  -group of order p3p3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence. It is well known that indecomposable summands in the complete stable splitting correspond to simple modules for the mod p double Burnside algebra. We shall use representation theory of the double Burnside algebra and the theory of biset functors.