Extra special defect groups of order p3 and exponent p2

In this paper we give results on (p-)blocks with the defect groups isomorphic to an extra special group of order p3. We are particularly interested in the number of irreducible ordinary characters, and the number of irreducible Brauer characters in the block. The situation splits naturally into two cases according to the exponent of the extra special group; in this paper we concentrate on the exponent p2 case. We prove that Olsson's Conjecture (Theorem 4.6) and Brauer's k(B)-Conjecture (Proposition 5.13) hold for the exponent p2 case. We are also able to calculate two important block invariants (k(B)−l(B) and l(B)−mG,B(1)(1)) in this case.