Donovan's conjecture and extensions by the centralizer of a defect group

We consider Donovan's conjecture in the context of blocks of groups $G$ with defect group $D$ and normal subgroups $N \lhd G$ such that $G=C_D(D\cap N)N$, extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form $Q_8 \times C_{2^n}$ or $Q_8 \times Q_8$ defined over a discrete valuation ring.


Introduction
Let p be a prime and k := F p . Let (K, O, k) be a p-modular system, so O is a complete discrete valuation ring with residue field k. Donovan's conjecture states that for a given finite p-group P , there are only finitely many Morita equivalence classes amongst blocks of finite groups with defect groups isomorphic to P (this may be stated over k or O). In reducing Donovan's conjecture to quasisimple groups, we must inevitably compare blocks B of finite groups G with those of normal subgroups N. The case where N contains a defect group D of B was treated by Külshammer in [25] for k-blocks, and by Eisele in [16] for O-blocks. This paper concerns the problematic case of normal subgroups of index a power of p, where it suffices to assume G = ND. The subcase that D is abelian was first considered, for k-blocks and with an additional splitting condition, in [23]. In [13] the full D abelian case was treated by introducing strong Frobenius numbers, related to the Morita Frobenius numbers introduced in [21]. The approach taken here also involves strong Frobenius numbers.
The purpose of this paper is to extend the reduction result above to the case that G = C D (D ∩ N)N. As an application, we show that Donovan's conjecture with respect to O holds when D ∼ = Q 8 × C 2 n or Q 8 × Q 8 for some n. Blocks with defect group Q 2 m × C 2 n were studied by Sambale in [29] and the number of irreducible characters and Brauer characters computed. Donovan's conjecture for O-blocks with defect group Q 8 was proved by Eisele in [15].
The structure of the paper is as follows. In Section 2 we treat strong Frobenius numbers and prove Theorem 1.2. We prove Theorem 1.3 in Section 3. In Section 4 we show that there are no blocks of quasisimple groups with defect group Q 8 × Q 8 , and few with defect group Q 8 × C 2 n for n ≥ 1. We give some preliminary reductions and results about blocks with the above defect groups in Section 5, and the proof that Donovan's conjecture holds for these blocks in Section 6.

Strong Frobenius numbers and extensions by the centralizer of a defect group
Throughout this section, let G be a finite group and B a block of OG with defect group D. We denote by Irr(G) the set of irreducible characters of G and Irr(B) the subset of Irr(G) of irreducible characters lying in the block B. We write kB for the block of kG corresponding to B. We denote by e B ∈ OG the block idempotent for B and by e χ ∈ Q c G the character idempotent for χ ∈ Irr(G), where Q c is the universal cyclotomic extension of Q. If A 1 and A 2 are finitely generated k-algebras (respectively O-algebras), we write A 1 ∼ Mor A 2 if A 1 and A 2 are Morita equivalent as k-algebras (respectively O-algebras). We quote the following definition from [13,Definition 3.2].
Definition 2.1 Let q be a, possibly zero or negative, power of p. We denote by − (q) : k → k the field automorphism given by λ → λ 1 q . Let A be a k-algebra. We define A (q) to be the k-algebra with the same underlying ring structure as A but with a new action of the scalars given by λ.a = λ (q) a, for all λ ∈ k and a ∈ A. For a ∈ A we define a (q) to be the element of A associated to a through the ring isomorphism between A and A (q) . Note that we have kG ∼ = kG (q) as we can identify − (q) : kG → kG with the ring isomorphism: From now on, we identify (kB) (q) with the image of kB under the above isomorphism. We define B (q) to be the unique block of OG satisfying k(B (q) ) = (kB) (q) . By an abuse of notation, we also use − (q) to denote the field automorphism of Q c defined by ω p ω p ′ → ω p ω 1 q p ′ , for all p th -power roots of unity ω p and p ′ th roots of unity ω p ′ and also the ring automorphism If χ ∈ Irr(G), then we define χ (q) ∈ Irr(G) to be given by χ (q) (g) = χ(g) (q −1 ) , for all g ∈ G. Note that if χ ∈ Irr(B), then (e χ ) (q) = e χ (q) and χ (q) ∈ Irr(B (q) ).
For R ∈ {Z, Q c , O}, we define the R-linear map ζ G : RG → RG, g → g p g p p ′ , where g p and g p ′ are the p-part and p ′ -part of g respectively. In general, the relevant ring R should always be clear from the context. The following was proved in [13,Proposition 3.5].
We now generalise the above Proposition to deal with normal subgroups of index a power of p. In what follows, A H will denote the set of fixed points in A under the action of H, where A is an algebra with an action of a group H. In practice, A will always be the group algebra of G or one of its blocks with the natural conjugation action of H ≤ G. For each n ∈ N, we use ω n ∈ Q c to denote some fixed primitive n th root of unity.

Proposition 2.3
Let N ✁ G be of index a power of p. Then ζ G induces a Z-algebra automorphism of (ZG) N .
Proof. As noted in the proof of [13,Proposition 3.5], ζ G commutes with conjugation by any g ∈ G. Therefore, since (ZG) N has a Z-basis consisting of the N-conjugacy class sums of G, ζ G maps (ZG) N to itself. Hence, it is sufficient to prove that ζ G induces a Q c -algebra automorphism of (Q c G) N ∼ = Q c ⊗ Z (ZG) N . We do this by showing that ζ G induces an isomorphism for all χ ∈ Irr(N). Note that Stab G (χ) = Stab G (χ (p) ) and if g / ∈ Stab G (χ), then C N g e χ = e χ C N g e χ = C N g e χ g e χ = 0, where C N g ∈ ZG is the sum of the elements in the N-conjugacy class containing g. We may, therefore, assume that G = Stab G (χ). For each g ∈ G we define e χ,gN := where χ ′ is an extension of χ to g N and λ is the linear character of g N given by If we define the Q c -algebra automorphism (1) Therefore, e χ,gN ∈ g(Q c Ne χ ). Since we are making a choice of extension χ ′ of χ, e χ,gN is only defined uniquely up to multiplication by some power of ω ord(gN ) . With this in mind, we introduce the notation of α ≈ β, for α, β ∈ (Q c G) N , if α = µβ for some p th -power root of unity µ ∈ Q c . Note that if when defining e χ,g −1 N we choose the same extension χ ′ of χ as when defining e χ,gN , i.e., Therefore, for any choice of e χ,g −1 N , e χ,gN e χ,g −1 N ≈ e χ and so we have shown that (Q c G) N e χ is a crossed product of G/N with Z(Q c Ne χ ) = Q c e χ in the sense of Külshammer [25]. In other words, (Q c G) N e χ = gN ∈G/N Q c e χ,gN . Now where the second equality follows from Proposition 2.2. Therefore, ζ G induces a Q cvector space isomorphism between (Q c G) N e χ and (Q c G) N e χ (p) . Before proceeding we note that, for all g, h ∈ G, (e χ,gN ) ord(gN ) = e χ . Also, as noted just after (1), e χ,gN = gxe χ , for some x ∈ (Q c Ne χ ) × and so e χ,gN e χ,hN e −1 χ,gN = ge χ,hN g −1 ≈ e χ, g hN . We use both these facts below in (3).
By (2), ζ G (e χ,gN ) = e χ,hN and so we need only show that ζ G (e χ,gN e χ,hN ) = (e χ,gN e χ,hN ) (p) . Since ζ G fixes coefficients that are p th -power roots of unity, it suffices in turn to prove that e χ,gN e χ,hN ≈ e χ,ghN . We prove this last statement via induction on the lowest layer of gN, hN or ghN in the upper central series of G/N. Note that e χ,gN e χ,hN ≈ e χ,ghN ⇔ e χ,hN e χ,(gh) −1 N ≈ e χ,g −1 N ⇔ e χ,(gh) −1 N e χ,gN ≈ e χ,h −1 N and so we may assume that hN is in the lowest layer. If h ∈ N, then e χ,hN = e χ and so e χ,gN e χ,hN = e χ,ghN . Now let g, h be arbitrary elements of G and set p n := max{ord(gN), ord(hN), ord(ghN)}. Then (e χ,gN e χ,hN ) p n ≈ (e χ,gN ) p n (e χ,h g p n −1 N . . . e χ,h g N e χ,hN ) ≈ (e χ,h g p n −1 N . . . e χ,h g N )(e χ,h −1 N ) p n −1 where the fourth and fifth relations follow from the inductive hypothesis. Now, since e χ,gN e χ,hN ∈ gh(Q c Ne χ ) ∩ (Q c G) N e χ = Q c e χ,ghN , we have e χ,gN e χ,hN ≈ e χ,ghN . ✷ For the following definitions see [21] and [13]. , for all χ ∈ Irr(B) or, equivalently, that φ restricted to Z(B) coincides with ζ •n G . Such an isomorphism φ is called a strong Frobenius isomorphism of degree n.
Before the next lemma we need to give a brief overview of Picard groups of blocks. For a more detailed discussion see [5, §1]. Let Proof. By [27,Corollary 5.5.6], B is the unique block of OG covering b and so e B is the sum of e b and its G-conjugates. However, [27,Theorem 5.5.10(v)] gives that b is G-stable, so that e B = e b and D ∩ N is a defect group for b.
Since B N is G/N-graded, it remains to find a unit a gN in the gN-graded component of B N , for each gN ∈ G/N. Let g ∈ C D (D ∩ N) and consider c g ∈ Aut(B) given by conjugation by g. Now c g induces the element In particular, M ∈ T (b) and so, by [5, Theorem 1.
. Therefore, by the comments preceding the lemma, iMi ∈ Out D∩N (ibi), where i is a source idempotent for b. Now by [28,14.5,Proposition 14.9], In particular, iMi has p ′ -order in Out(ibi) or, equivalently, M has p ′ -order in Out (b). However, since g ∈ D, c g has order a power of p meaning M induces the trivial auto-equivalence and c g ∈ Inn (b).
Let gN be a left coset of N in G, where we choose coset representative g ∈ C D (D ∩ N). Set α g ∈ b × such that c g is given by conjugation by α g . We set In the case D = (D ∩ N)Z(D), the proof of Theorem 1.2 can be greatly shortened as we may avoid reference to Proposition 2.3. Indeed, given the comments following Lemma 2.5, we only need that ζ G induces an O-algebra automorphism between Z(B) and Z(B (p) ). The proof then follows in very much the same vein as the abelian defect group case in [13,Theorems 3.16]. ✷

A reduction theorem for Donovan's conjecture
The following reduction result for Cartan invariants is presumably well-known, but we provide a proof. Write c(B) for the largest entry of the Cartan matrix of a block B. Proof.
(i) Let B ′ be the unique block of k Stab G (b) covering b. Then, by [24, Theorem C], B ′ ∼ Mor B and so we may assume that b is G-stable and, as noted in the proof of Lemma 2.5, that e B = e b . Therefore, by Green's Indecomposability Theorem,  Let G be a finite group and B a block of OG with defect group D isomorphic to a subgroup of P . We claim that c(B) ≤ |D|c and sf O (B) ≤ s. Suppose that c(B) > |D|c and that |G| is minimal with respect to these conditions. By the definition of the constant c, B is not in X and so there is a proper subgroup N ✁ G with contradicting the minimality of G. A similar argument using Theorem 1.2 shows the bound on strong O-Frobenius numbers. We have shown that the Cartan invariants and the strong O-Frobenius numbers of all blocks with defect group isomorphic to P are bounded, and so the result follows by [14,Corollary 3.11]. ✷ 4 Blocks of quasisimple groups with defect groups Q 8 × C 2 n and Q 8 × Q 8 Let G be a block of a finite group G with defect group D and maximal B-subpair We will use [31,Theorem 4.8] to observe that every block with defect group Q m 8 × A for m ≥ 0 and A an abelian 2-group is controlled, and apply the classification of controlled blocks of quasisimple groups given in [2]. To do so we first review some notation.
Let F be a saturated fusion system on a p-group D. A subgroup Q ≤ D is weakly F -closed if for any φ ∈ Hom F (Q, D) we have φ(Q) = Q, and Q is strongly F -closed if for any P ≤ Q and any φ ∈ Hom F (P, D) we have φ(P ) ≤ Q.
The normalizer N F (D) is the fusion subsystem of F on D such that for all P, Q ≤ D, the morphisms Hom N F (D) (P, Q) are those φ ∈ Hom F (P, Q) such that there isφ ∈ Hom F (D, D) extending φ.
A p-group D is called resistant if F = N F (D) whenever F is a saturated fusion system on D.
then G is a quotient of a classical group of Lie type not of type A or 2 A, defined over a field of order q a power of an odd prime and B corresponds to a non-quasi-isolated block of the corresponding group of Lie type. Furthermore, if m = 1 and A ∼ = C 4 , then G ∼ = Sp 2r (q) for some r.
Proof. By the discussion above, all blocks with these defect groups are controlled. The result follows directly from [2, Theorem 1.1] and its proof. ✷ When we come to consider blocks of arbitrary groups with defect group Q 8 × Q 8 , we must deal with the case of blocks b of Sp 2r (q) with defect group Q 8 × C 4 . We show that there can be no overgroup of Sp 2r (q) in Aut(Sp 2r (q)) possessing a block covering b with defect group Q 8 × Q 8 .
In what follows, for r ∈ N and q a power of a prime, we set I r ∈ GL r (q) to be the r × r identity matrix, and We define Proof. We first describe P ≤ H (see [18] for a description of defect groups of finite classical groups). We denote by i ∈ F × q a primitive 4 th root of unity. Then P = P 1 × P 2 ≤ Sp 2 (q) × Sp 2(r−1) (q) ≤ Sp 2r (q), where P 1 = Syl 2 (Sp 2 (q)) ∼ = Q 8 is generated by Since C G (H) ≤ H, G/H ֒→ Out(H) and as 8 ∤ (q − 1), q is not a square and so H has no field automorphisms of order 2. It follows that Out(H) has a normal Sylow 2-subgroup of order 2 generated by the diagonal automorphism induced by iI r 0 0 I r .
Suppose D ∼ = Q 8 × Q 8 . Then we may choose D such that the unique block of OHD covering b has defect group D. Moreover, the image of D in Out(H) is generated by the non-trivial diagonal automorphism. We will have reached our desired contradiction once we have proved that no h ∈ D\P commutes with P 1 . Since it is enough to show there exists no g ∈ Sp 2 (q) such that g 1 g −1 ∈ C CSp 2 (q) (P 1 ). This follows from the fact that C CSp 2 (q) (P 1 ) = Z(CSp 2 (q)) and that conjugation by g 1 is not an inner automorphism of Sp 2 (q). These two facts can be readily checked. ✷ 5 Blocks with defect group Q 8 × C 2 n or Q 8 × Q 8 We begin by gathering together some information on subgroups and automorphism groups of these 2-groups, easily verified by the reader. Blocks with defect group Q 8 ×C 2 n were studied in [29], where many of their numerical invariants were computed.
In particular, every subgroup of P has the form 1, C 2 m , C 2 × C 2 m , C 4 × C 2 m , Q 8 , Q 8 × C 2 m or C 4 ⋊ C 2 m for some m, with m ≥ 2 in the final case.
(b) The proper subgroups of Q 8 ×Q 8 are isomorphic to the following: 1, C 2 , C 4 , C 2 ×C 2 , The key to our treatment of blocks with defect group Q 8 × C 2 n or Q 8 × Q 8 is that in most cases covered blocks of normal subgroups of index 2 are nilpotent. This is covered in the following lemma.
Proof. Let D be a defect group for B. (i) Since D is abelian, it suffices to observe that Aut(D) is a 2-group. (ii) Since C 4 ⋊ C 2 n is metacyclic, by [10,Theorem 3.7] there is only one saturated fusion system on this 2-group, and so B must be nilpotent.
(iii) There is only one saturated fusion system on C 4 ⋊ Q 8 by [30, Table 13.1]. ✷ We summarize the results of [29] that we need here: Proposition 5.4 ( [29]) Let B be a block with defect group Q 8 ×C 2 n for some n. Then one of the following occurs: (i) k(B) = 2 n · 7 and l(B) = 3; (ii) B is nilpotent, k(B) = 2 n · 5 and l(B) = 1.
Proof. This follows from [29, Lemma 2.2, Theorem 2.7]. ✷ The next result will be used frequently without reference throughout the remainder of the article. Recall that a p-solvable group G has p-length one if there are normal subgroups N, M ✁ G, with N ≤ M, such that N and G/M are p ′ -groups and M/N is a p-group. The abelian Sylow 2-subgroup case of the following is well-known, and possibly also the Q 8 case, but since we do not know a reference we include a proof. Lemma 5.9 Let G be a solvable group with Sylow 2-subgroups which are abelian or Q 8 . Then G has 2-length one. If further G has cyclic 2-subgroups, then it is 2-nilpotent.
Proof. We may assume that O 2 ′ (G) = 1, so that O 2 (G) = 1 and C G (O 2 (G)) ≤ O 2 (G). Let P ∈ Syl 2 (G). If P is abelian, then P ≤ C G (O 2 (G)) ≤ O 2 (G) and we are done. Suppose that P ∼ = Q 8 . Since Z(P ) is the unique subgroup of P of order two, we have Z(P ) ≤ O 2 (G). If O 2 (G) = Z(P ) or O 2 (G) = P , then we are done. If O 2 (G) ∼ = C 4 , then G/O 2 (G) is a 2-group and again we are done.
In the case that P is cyclic, the fact that G is 2-nilpotent follows from Aut(P ) being a 2-group. ✷ The following is by now a standard reduction when treating Donovan's conjecture, using Fong reductions and [26]. the two components). By considering all of the possible expressions of Q 8 × Q 8 as a central product of two groups in Lemma 5.1, we must have (without loss of generality) D 1 and D 2 ∼ = Q 8 or Q 8 × C 2 , otherwise B covers a block of a component with cyclic defect group, which forces a contradiction as in the previous paragraph. Note that O 2 (G) ≤ Z(G) in this case. We are now in case (b) Now suppose that t = 1, so F * (G) = O 2 (G)Z(G)L 1 . Since, by Lemma 5.3, every block with defect group C 4 ⋊ Q 8 , C 4 ⋊ C 2 m or C 4 × C 2 s is nilpotent for m ≥ 2 and s = 2, we may assume, by Lemma 5.1, that D 1 ∼ = Q 8 × Q 8 , Q 8 × C 2 m , Q 8 , C 4 × C 4 or C 2 × C 2 for some m ≥ 1. We treat each of these cases in turn.
By Proposition 4.2 we cannot have D 1 ∼ = Q 8 × Q 8 . Suppose that D 1 ∼ = Q 8 × C 2 m for m ≥ 1. Consider first the case D ∼ = Q 8 × C 2 n . By Proposition 4.2 L 1 is of classical type other than A or 2 A as in case (a)(ii) of the statement with N = L 1 . That G/N has cyclic Sylow 2-subgroup ND/N is immediate, and so by Lemma 5.9 G/N is 2-nilpotent. Considering the outer automorphism groups of such quasisimple groups we have that O 2 ′ (G/L 1 ) is supersolvable as required. As a note to this last calculation, observe that unless L 1 has type D 4 the only odd order elements of the outer automorphism group are field automorphisms, so that O 2 ′ (G/L 1 ) is cyclic. In the case of type D 4 we may either analyse [2, Theorem 1.1] a little more deeply and observe that this case does not after all occur, or observe that a Hall 2 ′ - Finally we claim that we cannot have D 1 ∼ = C 4 × C 4 or C 2 × C 2 . Suppose that we do and so D/D 1 is an abelian. Let M be the preimage in G of O 2 ′ (G/L 1 ), and let B M be the unique block of OM covered by B. Then B M has defect group D 1 . Now there is a subgroup M 1 of G containing M with [M 1 : M] = 2. The unique block , since it is a subgroup of D. In the Q 8 × C 4 case we obtain a contradiction by Lemma 5.7. In the C 4 ⋊ Q 8 and C 4 × C 2 cases by Lemma 5.3 B M , and so by Proposition 5.5 b i , must be nilpotent, a contradiction.  Table 5]) we see that every normal subgroup of O 2 (G) is also normal in G, so B ′ is quasiprimitive. Hence we see that B ′ is also a reduced block satisfying (a)(ii). We may now assume that O 2 (G) = G. Now let B N be the unique block of the quasisimple group N covered by B. By Lemma 5.6 B is Morita equivalent to B N , and note that they share a defect group. Hence we may assume that G = N since Morita equivalence preserves both of these invariants (see [13,Proposition 3.12] for the latter).
We make use of [7], to which we refer for notation. Assume for the moment that G is a group of Lie type, i.e, the centre is largest possible. We note that Z(G) is a 2-group for the groups we are considering. Here the identity element is the only quasi-isolated element (see for example [6, Table 2]) and so the principal block is the only quasiisolated block. However, for groups of these types the principal block of G (or that of any quotient of G) cannot have the given defect groups, so we may assume our block is not quasi-isolated. We may now apply the Bonnafé-Dat-Rouquier correspondence [7,Theorem 7.7], so that B is Morita equivalent to a block C 1 (with isomorphic defect group) of a proper subgroup H 1 of G. We note that the well-known error in [7] does not apply in our situation, since we are working with 2-blocks and the centre of G is a 2-group. If G is not a group of Lie type (i.e., the centre is not largest possible), then we note that by, for example, [12,Proposition 4.1] the Bonnafé-Dat-Rouquier correspondence induces a Morita equivalence modulo central 2-subgroups and we may apply the same argument.
Applying Proposition 5.10 to C 1 , it is Morita equivalent to a block C 2 in a reduced pair (H 2 , C 2 ) with isomorphic defect groups, where [H 2 : We have c(B) = c(C 1 ) and sf O (B) = sf O (C 1 ). Now apply Proposition 5.11 to (H 2 , C 2 ). Either we are in case (a)(i) of Proposition 5.11 or we may repeat the above argument. Since the index of the 2 ′ -part of the centre strictly decreases each time we apply the Bonnafé-Dat-Rouquier correspondence, repetition of this process must eventually end in case (a)(i) of Proposition 5.11. Now let (G, B) be a reduced pair satisfying condition (a)(i), so there is N ✁ G such that G = ND and G ∩ N ∼ = Q 8 . Let b be the unique block of ON covered by B, noting that this has defect group Q 8 . By [15] there is a unique Morita equivalence class of blocks with defect group Q 8 and a given Cartan matrix, so mf O (b) = 1. By Theorem 1.2 and [13, Corollary 3.11] sf O (B) ≤ sf O (b) ≤ |Q 8 | 2 !. Considering Cartan invariants, c(b) ≤ 8 and so, by Lemma 3.1(i), c(B) ≤ 2 n+3 . ✷ We remark that we cannot at present so easily obtain a similar bound on the strong O-Frobenius number for blocks with defect group Q 8 × Q 8 , since we do not know how this invariant behaves with respect to normal subgroups of p ′ -index.
We further remark that in order to bound only the strong Frobenius number of a quasisimple group we could have used [17].
6 Donovan's conjecture for blocks with defect group Q 8 × C 2 n or Q 8 × Q 8 We are now in a position to verify Donovan's conjecture for Q 8 × C 2 n and Q 8 × Q 8 .
Proof of Theorem 1.4. By Proposition 5.10, in verifying Donovan's conjecture it suffices to consider reduced blocks.
Consider first Q 8 × C 2 n . From [14,Corollary 3.11] we need only bound strong O-Frobenius numbers and Cartan invariants for reduced blocks with defect groups isomorphic to Q 8 × C 2 n , hence the result follows in this case by Corollary 5.12. Now consider blocks with defect groups Q 8 × Q 8 . By Proposition 5.11 either: (i) there are normal subgroups N ✁ H ✁ G with H = ND and D ∩ N ∼ = Q 8 or Q 8 × C 2 , and [G : H] odd; or (ii) there are commuting N 1 , N 2 ✁ G with N 1 ∩ N 2 ≤ Z(G) such that D ∩ N 1 , D ∩ N 2 ∼ = Q 8 or Q 8 × C 2 , and [G : N 1 N 2 ] is odd.
By [16,Corollary 4.18] it suffices to show that there are only finitely many possibilities for the Morita equivalence class of the unique block of OH or ON 1 N 2 covered by B. Hence we may assume that G = H in case (i) and G = N 1 N 2 in case (ii).
As above, from [14, Corollary 3.11] we need only bound strong O-Frobenius numbers and Cartan invariants for such blocks.
In case (i) we have sf O (B) ≤ 16 2 ! and c(B) ≤ 2 6 using arguments as in Corollary 5.12, noting that D ∼ = Q 8 × Q 8 with D ∩ N ∼ = Q 8 or Q 8 × C 2 satisfies the conditions of Theorem 1.2. Suppose that we are in case (ii). Note that G ∼ = (N 1 × N 2 )/W for some group W ≤ Z(N 1 × N 2 ). Now B corresponds to a block A of N 1 × N 2 with O 2 ′ (W ) in its kernel and defect group Q 8 × Q 8 , Q 8 × Q 8 × C 2 or Q 8 × Q 8 × C 2 × C 2 . Write A i for the block of N i covered by A, with defect group Q 8 or Q 8 ×  (5)) and O(SL 2 (5)×SL 2 (5)), and a non-principal block of (Q 8 × Q 8 ) ⋊ 3 1+2 + , where the centre of 3 1+2