Structure of blocks with normal defect and abelian p′ inertial quotient

Let k be an algebraically closed field of prime characteristic p. Let kGe be a block of a group algebra of a finite group G, with normal defect group P and abelian p' inertial quotient L. Then we show that kGe is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order p³ with a quaternion group of order eight with the centre acting trivially. In the case of p = 3, we give explicit generators and relations for the basic algebra as a quantised version of kP. As a second example, we give explicit generators and relations in the case of a group of shape 2¹+⁴ : 3¹+² in characteristic two.