NEW ZEALAND JOURNAL OF MATHEMATICS Volume 25 (1996), 107-131
DADE’S CONJECTURE FOR THE TITS GROUP
JlANBEI A n
(Received November 1995)
Abstract. This paper is part of a program to study the conjecture of Dade on counting characters in blocks for several finite groups. The local structures of certain radical chains of the Tits group are given, and Dade’s conjecture and Alperin’s weight conjecture are confirmed for this group.
Introduction
Let G be a finite group, p a prime, and B a p-block of G. In his papers [7, 8, 9] Dade has presented a conjecture expressing the number of ordinary irreducible characters with a fixed height in B in terms of the numbers of ordinary irreducible characters of related heights in related p-blocks B' of certain local subgroups of G. By Dade [9], the final conjecture (inductive conjecture) can be proved by verifying it for all non-abelian simple groups. So far it has been verified for all Mathieu groups, Ji, J2, J3, L<2 {q), Sz(q), and 2G2(<72) in non-defining characteristics. In this paper we confirm the final conjecture for the simple Tits group. By Dade [9], the final form is equivalent to the invariant form whenever a finite group has a trivial Schur multiplier and an outer automorphism group all of whose Sylow subgroups are cyclic. Thus it is suffices to confirm Dade’s invariant conjecture for the Tits group, since the group has a trivial Schur multiplier and a cyclic outer automorphism group of order 2 (cf. [6 , page 74]). Let ii be a p-subgroup of G and if an ordinary irreducible character of the normalizer N g {R) of R in G. Then (R,if) is called a B-weight of G if if is of p-defect 0 as a character of N g {R)/R and the p-block B( 1991 AMS Mathematics Subject Classification: 20C20. 108 JIANBEI AN 1. The Invariant Conjecture of Dade Throughout this paper we shall follow the notation of Dade [7]. Let G be a finite group and Op(G) the maximal normal p-subgroup of G, where p is a prime. A chain of p-subgroups of G C : P0 < Pi < ••• < Pn, (1.1) is called a radical p-chain if P0 = Op(G) and Pi = Op(r\j=0NG(Pj)) for all t with 1 < t < n. If C is such a chain, we denote by N g {C ) the normalizer fl^= 1 N c{P j) of C in G, and by \C\ the length n of C. We define the centralizer C g (C ) of C in G to be the centralizer Cg{Pti) of Pn. A p-subgroup R of G is called a radical p-subgroup of G if R = Op(Ng(R))- Thus the p-chain C given by ( 1.1) is a, radical p-chain if and only if Po = Op(G) and Pp is a radical p-subgroup of n*~o7Vc?(Pj) for I > 1. Denote by 7Z = 1Z(G) the set of all radical p—chains of G. Then G acts on TZ by conjugation. Let Irr(G) be the set of all ordinary irreducible characters of G, and Irr(P) the set of irreducible characters of G which belong to B in the usual sense. Let d(J9) be the defect of a block B 6 Blk(G), so that \D\ = pd(fi) for any defect group D of B. Given a p-block B and a radical p-chain G, let N (C ) = g N{C), and let Blk (N(C) | B ) be the set of p-blocks b of N(C) such that B = bG, that is, such that B corresponds to b under the Brauer correspondence. Given a positive integer n, let a(n) be the unique non-negative integer such that pa(n) jg thg largest power of p (exactly) dividing n. Denote by a(G) the integer a(|G|) when G is a finite group. For each x € Irr(G), we write d(x) for the p-defect of X) the unique non-negative integer such that a (x (l)) = a (G )-d (x ). (1.2) Suppose the center Z(G) of G is trivial. Then we can identify G with its inner automorphism group Inn(G), so that G is a normal subgroup of its automorphism group A — Aut(G), and the outer automorphism group O = Out(G) is the factor group A/G. In addition, A acts naturally on each p-subgroup chain G, and the stabilizer of G under this action is denoted by Na(C). Thus Ng{C) = G f l Na(C) and Ng(C) < Na{C). The factor group Na(C)/ Ng{C) is isomorphic to the sub group No(C) = Na{C)G/G of O. In addition, the image of a radical p-chain under the action of an element of A is also a radical p-chain, so A also acts on the set 71(G). Moreover, Na(C) acts on each character -0 of Irr (Ng(C)), and the stabilizer Na(C,'ip) of ip in Na(C) is a subgroup of Na(C) containing Ng(C). Thus Na(C,iP)/Ng(C) is isomorphic to the subgroup No(C,ip ) = Na{C,iP)G/G of N0 (C). Suppose G e Tt(G) and B a p-block of G, and suppose d is a non-negative integer and U is a subgroup of O = Out(G). Let Irr(C,B ,d ,U ) = {V> € Irr {Na (C)) : B ( i p f = B, d«>) = d ,N 0 (C,i>) = U) (1.3) and define k(C,B,d,U) = \ Irr(G, B, d, U)\. Since Z(G ) = 1, the invariant conjec ture of Dade [9, 2.5] is stated as follows. DADE’S CONJECTURE FOR THE TITS GROUP 109 Dade’s Invariant Conjecture. If Op(G) = 1 and B is a p-block of G with defect d(B) > 0, then for any integer d > 0 and any subgroup U < Out(G), £ (-l)'clk(C,S, d,U) = 0, (1.4) cen /G where 1Z/G is a set of representatives for the G-orbits in 1Z. From [9] the invariant conjecture is equivalent to the final conjecture whenever G has a trivial Schur multiplier Mult(G) and an outer automorphism group all of whose Sylow subgroups axe cyclic. If O = Out(G) is cyclic, then U is determined by its order w = \U\, so that the equation (1.4) is equivalent to the following equation £ (—l) |c|k(C,B,d,w) = 0, (1.5) C£K/G where k(C", B, d, \U\) = k(C", B, d, U ) for any p-chain C", and w is a factor of the order |0 |. 2. Radical p-Chains of the Tits Group The notation and terminology of Section 1 are continued in this section. We shall also follow the notation of [6 ] and [15]. In particular, given a sign r], if p = 2, then we shall use 2*+27 to denote the extraspecial 2-group of order 21+27 and type r], and if p is odd, we use p*+27 to denote the extraspecial p-group of order p1+27 with exponent p or p2 according as 77 = + or —. If X and Y are groups, we use X . Y to denote an extension of X by Y, and X : Y to denote a split extension of X by y . Also given a positive integer n, we use E pn or simply pn to denote the elementary abelian group of order pn and [n] to denote an arbitrary group of order n. We use Z n or simply n to denote the cyclic group of order n and £>2n to denote the dihedral group of order 2n. Let G be the simple Tits group and A = Aut(G). Then A is the Ree group 2F4(2) and the Schur multiplier Mult(G) is trivial. We may suppose G is a subgroup of A , so G is the commutator subgroup of A and A/G ~ Z 2. Suppose r € A\G, so that t 2 € G and A = (G ,r ). The order |G| of G is |G| = 211 • 33 • 5 2 • 13, so that a Sylow 13-subgroup is cyclic and the conjecture for p = 13 follows by Dade [10]. Thus we may suppose p € {2,3,5}. Suppose C G TZ{G) is given by ( 1.1) with \C\ > 1. Then Pi is a radical p-subgroup of G, so that we need first to classify radical p-subgroups of G. Let <3>(G,p) be a set of representatives for conjugacy classes of radical p-subgroups of G. If p = 5, then we may take *(G ,5 ) = { 1,^ 25 }, (2.1) since Gg(5^4) ~ Z5 x Dio (cf. [15, page 550]). Moreover, Gg(-^ 25 ) = -^25 , and by [15, Theorem 1 and Corollary], N g {E 2 5 ) = E 25 : 4. A 4 and N a{E 25 ) = -E25 : 4. S4, where S4 and A 4 axe symmetric and alternating groups on 4 letters, respectively. Let C (l) and G(2) be radical 5-chains as follows: C ( l ) : 1 C(2) : 1 < E25. (2.2) 1 1 0 JIANBEI AN Then we may suppose 1Z(G)/G = {(7(1), £ (2 )}. If p = 3, then by [15, page 550], there are two classes of elementary abelian subgroups of order 9 in G, each of which has normalizer isomorphic to Eg : GL2 (3) ~ 32 : 2 . 5 4 . Thus G has two classes (with representatives Eg and Eg, respectively) of radical 3-subgroups of order 9. Moreover, C g {E 9) = C Na(Eg)(Eg) = Eg and Cg{Eq) — Eg. Let S = 3++2 be a Sylow 3-subgroup of G. We may take $(G , 3) = {1 ,B 9)£ 9*,3V+2}. (2.3) Suppose 3+ contains Eg and Eg. Define the radical 3-chains as follows: C{ 1) : 1 C{2) 1 < Eg (7(3) : 1 < Eg < 3i+2 <7(4) 1 < m <7(5) : 1 < Eg < 3i+2 <7(6) 1 < 3i_+2. (2.4) Then we may suppose TZ(G)/G — { C (i) : i = 1 ,... ,6}. Thus we have shown the first and second part of the following proposition. (2A ). Let G be the Tits group and A = Aut(<7), so that A = (G ,t) for some t £ A\G. Suppose p £ {3,5}. (a) We may suppose = f{C(l),C(2)} ifp = 5, 11(G )/G = \{C{*) : * = 1,— ,6} i f p = 3, where each C (i) is defined by (2.2) or (2.4) according as p = 5 or 3. (b) Suppose p = 5 and C = C(2) defined by (2.2). Then C g{C ) — C g (E 2s) = E 2 5 , Ng(C) = E 25 '• 4. A 4 , and Na(C) = E 25 : 4. S4 . (c) Suppose p = 3 and C is a chain defined by (2.4) with n = |C| ^ 0. Then CG(C) = CG(Pn) = Z(P n) and I Eg : GL2(3) ifC = C{2)orC{4), N a (C) ~ I -.Et if C = C(3) or <7(5), 3'+ 2 : Ds if C = C(6), where Pn is the final subgroup of the chain C . We may suppose C (2)T = C ( 4), C(3)T = C(5). Moreover, Na(C) = N g(C ) except when C = C {6), in which case Na(C) = (3^.+2 : Ds,t) = {SU3(2), cr) and a £ A induces the inverse- transpose automorphism on SU3(2). Proof of (c). Since Eg is a maximal toms of A — 2F4(2) (cf. [13, Proposition 1.2 (2)]), E 9 and £ 9* are conjugate inA, and we may suppose Eg = Eg, so that C(4) = C (2)T. Similarly, we may suppose C(3)r = C(5). Thus ArA((7(i)) = NG(C(i)) for i £ {2,3,4,5}. By [13, Proposition 1.2], Na(3A) = NA(3]+2) ~ SU3(2) : 2 , and we may suppose N a (C (6)) = (SU3(2),cr), where a induces the inverse-transpose automorphism on SUs(2). It is clear that Cq{C) = Cg{Pti) = Z(Pn). If \C\ = 1, then Ng(C) = Ng(Pi) — Eg : GL2(3) or NG(C) = NG{S\+2) = 3^+2 : D 8 according as C £ {(7(2), C (4)} or C — (7(6). If C = (7(3) or (7(5), then N q(C ) is given above. This proves (2A). DADE’S CONJECTURE FOR THE TITS GROUP 1 1 1 If p = 2, then let Mi and M2 be maximal subgroups of G such that Mi ~ Z2 . [28] : E 5 : Z4 and M2 ~ E 4 . [28] : S3. Thus 0 2(M i) = Z2 . [28] and 0 2(M2) = E ±. [28] are radical 2-subgroups of G. Now G has three conjugacy classes of non-trivial radical 2-subgroups with orders 29, 210 and 211, respectively. It follows that we may take 4>(G,2) = {1 ,Z 2 .[28],,E4.|28],S }, (2.5) where S' is a Sylow 2-subgroup of G. For i e {1,2,3,4}, we define radical 2-chains C (i) of G as follows: G( 1) : 1 G(2) : 1 < Z2 . [28] C{3) : 1 < Z2 . [28] < S C (4) : 1 < EA . [28]. (2.6) Let 1Z°(G) be a G-invariant subfamily of 7Z(G) such that H °(G )/G = {C (1),C (2),C (3),C (4)}. (2.7) We have the following proposition. (2B ). Let p — 2 , and let G be the Tits group and A = Aut(G), so that A = (G, r) for some r G A\G. (a) If B is a 2-blocks with a positive defect, and if d and w are non-negative integers, then Y, ( - l ) |C|k (C,B,d,w) = Y, ( - l ) |C|k (C ,B ,d ,w ), C£lZ(G)/G ce7i°(G)/G where 1Z°(G) is given by (2.7). (b) Suppose C is a chain defined by (2.6) with n — \C\ ^ 0. Then C g (C) = CciPn) = Z(Pn), and 'Z2.[28]:£ 5:Z 4 if C = C {2), N a (C ) ^ Is if C = G(3), E 4 .[2 8]:S 3 i f C = C( 4), where Pn is the final subgroup of the chain C . In addition, we may sup pose Na{C) = (Ng(C), t). Moreover, NA(C{ 2)) ~ Z2 . [29] : E 5 : Z4 and N A ( C ( 4 ) ) ~ E 4 .[2 9} : S 3. P roof. (a) Suppose E± . [28] is a subgroup of S. Let G : 1 < E 4 . [28] < S and C ' : 1 < S. Then G and C" are radical 2-chains, |G| = 1+|G'| and N g {C ) = Ng{C') = S. Thus (—l)lc l k(G, B , d, w) + (—l)^c ' k(G', B, d, w) = 0 for any integers d, w > 0 and all 2-blocks B with a positive defect. A radical 2-subgroup of iVG(Z2 . [28]) properly containing Z 2 . [28] is a Sylow 2-subgroup of G. This proves (a). 1 1 2 JIANBEI AN (b) Since CG(Pn) is a 2-subgroup, CG(C) = Cg(Pti) = Z(Pn). If \C\ = 1, then the structures of N g (C ) follows by N g (C ) = N g{P i) and the proof above. Since NG{S) = S, it follows that Ng(C(3)) = S. By [6, page 74], JVa(Z2 . [28]) = Z2 . [29] : E b : Z4 and N A (E4 .[ 28]) = E 4 .[ 29] : S3. This proves (b). □ 3. Dade’s Conjecture for the Tits Group The notation and terminology of Sections 1 and 2 are continued in this section. (3 A ). Let G be the Tits group. If p € {2,3,5}, then G has exactly one p-block, the principal block Bo = B q(G), with a positive defect. Let d and w be non-negative integers. (a) Suppose p = 5. Then 8 if d = 2 and w = 2, k(G, Bo, d, w) = < 8 if d = 2 and w = 1, 0 otherwise. (b) Suppose p = 3. Then 3 if d = 3 and w = 2, 6 if d = 3 and u; = 1, k (G,B0,d,w) = 2 if d = 2 and w = 2 , 2 if d — 2 and w = 1, 0 otherwise. (c) Suppose p = 2. Then 8 z/ d = 11 and w = 2 , 2 if d = 10 and w — 2, 4 */ d = 10 and w = 1, 1 if d = 9 and w = 2, ]t(G,Bo,d,w) = < 2 if d = 9 and w = 1, 2 if d = 7 and w = 1, 1 if d = 5 and w = 2, 0 otherwise. (d) Suppose £(Bo) is the number of irreducible Brauer characters in B q. Then 14 if p — 5, ^(i?o) = < 9 if p = 3, 3 if p = 2. P roof. Let £? be a p-block with a positive defect, and D a defect group of B. Then we may suppose D € $(G ,p). By (2A) and (2B), Cg(-D) = Z(D) is a p-group. So a root block b of B is the principal block, and by Brauer’s Third Main Theorem, B is the principal p-block B q, where a root block of B is a p-block b of C g (D)D such DADE’S CONJECTURE FOR THE TITS GROUP 113 that b ° = B. Thus Irr(B) = Irr(G)\Irr°(G), where Irr°(G) is the set of characters of Irr(G) of p-defect 0. The proof of (a)-(c) follows by [6 , page 75]. (d) By a result of Brauer, the number of irreducible Brauer characters of G is the number t of conjugacy p'-classes of G. Thus ({Bo) = f-|Irr°(G)|, and (d) follows. □ (3B ). Let B be a p-block of the simple Tits group G with defect d(B) > 1. If p is odd, then B satisfies the invariant conjecture of Dade. P roof. By Dade [10], we may suppose a defect group D of B is non-cyclic, so that p e {5,3} and B = So = Bo{G). Thus a defect group D of B is a Sylow p-subgroup. If C is a radical p-subgroup of G, then by Brauer’s Third Main Theorem, Blk ( Nq{C) | B q) contains only the principal p-block of Nq(C). Using CAYLEY, we can calculate irreducible characters of the subgroup Ng{C)-, see the character table of Ng(C) in the Appendix. If p = 3, we need only the degrees of irreducible characters of Ng{C), and these are listed in the Appendix. Let A = Aut(G), and let Bo (A) be the principal block of A. Then D is a defect group of Bo(A). Suppose A = (G ,r) for some r e A \G , so that r 2 € G. If p = 5, then we may suppose D = E 25 6 $ (G , 5). Let C = (7(2) be given by ( 2 .2), N(C) = N g(C ), and let bo and 60 (A) be the principal blocks of N(C) and N a(C ), respectively. Then D is a defect group of both bo and 60 (A). By [14, Korollar], the Alperin-McKay and Brauer height conjectures both have an affirmative answer for Bq(A), so k ( B 0 (A)) =k(B o(A),0) =k(4o(A),0), where for a p-block b, k(6 ) = |Irr(6 )| and k(b,h) is the number of characters of Irr(6 ) with height h (or with p-defect d(6 ) — h). Since d (60 (A)) = 2, it follows by a result of Brauer and Feit [11, Theorem IV.4.18] that each character of Irr (60 (A)) has height 0, so that k (60 (A)) = k (60 (A ),0). Thus k (60 (A)) = k [B 0 (A)). By (3A) (a), Irr(Bo) has exactly 8 characters stabilized by r. Since B 0 (A) is the only 5-block of A covering Bo, it follows by Clifford theory that k (S 0(A)) = 16+4 = 20, so that k (60 (A)) = 20. On the other hand, by Table 4 in the Appendix N(C) has 16 irreducible charac ters, all of height 0. Since C g {D) = D, it follows that Irr (iV(C')) = Irr(6 o). Sup pose x is the number of characters of Irr( 6 o) stabilized by 2x + ^(16 — x) = k (bo(A)) - 20, and x = 8. Since Blk (N(C ) | Bo) = { 60 }, it follows that 8 if d = 2 and w = 2 , k (N (C (2)), Bo, d, w) = * 8 if d = 2 and w = 1, (3.1) 0 otherwise. Thus (3B) follows by (3A) (a) and (3.1). 114 JIANBEI AN Suppose p = 3. Let C = C (2) and C' = C(3). If b0 = B 0 (N{C)) and b'0 = B0 (N(C')), then Blk (N (C ) \ B 0) = {60} and Blk (N{C') \ B0) = {b'0}. A similar proof to above shows that Irr (iV(C')) = Irr(&o) and Irr (N(C')) = Irr(6g). By Tables 2 and 3 in the Appendix, N(C ) and N(C') both have 9 irreducible characters of 3-defect 3 and two of 3-defect 2. By (2A) (c), NA(C ) = N(C) and N a (C ) = N(C'), so that t does not stabilizes any character of Irr(6o) and Irr(6g). Thus 9 if d = 3 and w = 1, k (N {C ),B 0 ,d,w) = k{C',B 0 ,d ,w ) = < 2 if d = 2 and w = 1, (3.2) 0 otherwise. It follows that (3.3) By (2A) (c), C (2)T = C(4) and C(3)T = 0(5), and so JVG(C (2))T = A’6.(C(4)) and Ng(C(S))t = NG(C(o)). Thus (3.3) still holds for C = C(4j and C' = C(5). Thus it suffices to show the following equation: k ( N ( C { l) ) ,B 0 ,d,w) = k(C(6),B 0 ,d ,w ) (3.4) for all integers d,w > 0. Let C — C (6), and let bo and bo(A) be the principal 3-blocks of N q (C ) and N a (C), respectively. By (2A) (c), NG(C ) ~ 3^+2 : D 8 and NA(C ) ~ (SU3(2),a), where a acts as the inverse-transpose map on SUs(2). By [2, (3.6)], Irr (&o(^4.)) has 9 characters of 3-defect 3 and 5 characters of 3-defect 2. A proof similar to above shows that N g {C) has only one 3-block, so that Irr (N g (C)) = Irr(&o)- By Table 1 in the Appendix, Irr(&o) has 9 characters of 3-defect 3, and 4 characters of 3-defect 2. Let x be the number of characters x °f Irr(60) such that x T = X and d(x) = 3. By Clifford theory, 2x + |(9 — x) = 9 , and so x = 3. Similarly, if y is the number of characters x °f Irr(M such that XT = X and d(x) = 2, then 2y + 4(4 — y) = 5 and y — 2. Thus '3 if d = 3 and w = 2, 6 if d = 3 and w = 1, 2 if d = 2 and w = 2, (3.5) 2 if d = 2 and w = 1, 0 otherwise. So (3.4) follows by (3A) (b) and (3.5), and this proves (3B). □ (3C). Let B be a 2-block of the simple Tits group G with defect d(B) > 1. Then B satisfies the invariant conjecture of Dade. Proof. Let C be a 2-chain given by (2.6) with \C\ ^ 0 and let N (C ) = Ng {C). By (2B) (b), Na(C) = (N(C),t) for some r £ NA{C)\N{C). Thus r 2 € N(C) and A — (G, r). DADE’S CONJECTURE FOR THE TITS GROUP 115 (1) If C = G(2), then N(C) ~ Z2 . [28] : E5 : Z4 , and the ordinary character table of N(C) is given by Table 5 in the Appendix and we shall follow its notation. A proof similar to above shows that Irr(&o) = Irr (N(C)), where &o = Bq(N(C)). Suppose g and g' are elements ofN(C). If gT and g' are conjugate in N(C), then g and g' have the same order, and they have the same number of conjugates in N(C). In the notation of Table 5 the conjugacy classes 1 to 4, 8, 17 and 24 are stabilized by r. By Table 5, N(C ) has exactly three classes (classes 9,10,11) of elementsx such that |x| = 4 and x has 160 conjugates in N(C). But the centralizers of these three classes in G have different orders, so that they have different types in G. Thus r stabilizes each of these three classes. Since£5 = £5, it follows that £5(<7T) = £5(0) for all g G N(C). Thus the class 5 is stabilized by r. Similarly, it follows from the values of £25 £3 and £4 on the class 10 that £3 = £3, and so also the class 16 is stabilized by r. Suppose £ and £' are two characters of Irr (iV(C')). If £T = £', then £(gr)= £'(#) for all g G N(C). Thus £(<7) = £'(<7) when gT and g are conjugate in N(C). In particular, £ and £' have the same degree. Using the classes stabilized by r and the facts given above, we can get the following characters of Irr N(C( )) stabilized by r : {& :k € {1,3,5,6,7,18,19,26,27,28}}; and moreover, r stabilizes each of the following pairs of characters: (£2, £4 ), (£s, £9), (£10,£17), (£12,£14), (£20, 621), (£22, 623), (£24,£25)- Using CAYLEY, we can induce each character of Irr N(C( )) to G and decompose each induced character as a linear combination of characters in Irr(G). The decom positions are given by Tables 8 to 10 in the Appendix. By the Frobenius reciprocity theorem, (x,Ind®(c)(£)) = { x \ n ( C ) ,0 for each x G Irr(G) and £ G Irr (iV(C)), where x\n(C) is the restriction of x to N(C ), Ind^(C)(£) is the induced character by £, and (*, *) is the inner product of characters. Thus we can get the multiplicity of £ in x\n(C)- I f X T = X ' € Irr(G), then (x 'U (C ),D = ( x \ n ( C ) ,0 and we can get the multiplicity of £T in x'\n(C )- In the notation of [6, page 75], suppose x — X i 6 and x ' = X i7 - Then x T = x'- By Table 8 of the Appendix, (xliv(c), £24) = (x^ liv(c'), £25) = 2 and (x/ |at(C),£24) = (xIn(C),£25) = 3. Thus £24 = £25- Similarly, (xU(C),£22) = 0 and It follows that £[0 = £17, £12 = £14, £J4 = £25, and r stabilizes the others. So 8 if d = 11 and w = 2, 6 if d = 10 and w = 2, 4 if d = 10 and w = 1, 1 if d — 9 and w = 2, k(N(C(2)),B 0 ,d ,w ) = 2 if d = 9 and 10 = 1, (3.6) 2 if d = 8 and w = 2, 4 if d = 7 and w = 2, 1 if d = 5 and w = 2, 0 otherwise. (2) Suppose C = C(3), so that -/V’(C) ~ Z2 . [28] . Z4 is a Sylow 2-subgroup of G and its ordinary character table is given by Table 6 . A proof similar to above shows that Irr(&o) = Irr (N(C )). The calculation of the numbers k(C, B o,d,w ) for C is also similar to that for C(2) above. By Table 6 , r stabilizes the following characters of Irr(JV(C)): {& : fc € {1,4,12,15,21,22,23,25,26,31,35}}; and r permutes each of the following pairs of characters: (£2 , £5 ), (^3,^6 )? (£7, £s), (£9,£17), (£i3,£i8), (£19,£20), (£24,£27) and (£28,£29)- By Table 9, £9 = £17, £[3 = £18, £[g = £20, and r stabilizes the other characters except possibly the three pairs (£10,6 4 ), (£24,£27), and (£32,£34), in which case (xk(C ),£i) = (xIjv(C),&) for all X G Irr(G), where ( i j ) e {(10,14), (24,27), (32,34)}. However, the values of £15 and £29 on the class 17 in Table 6 imply that this class is stabilized by r, and this implies that £24 and £27 are stabilized by r. Similarly, the values of £16 and £30 imply that the class 19 is stabilized by r, and the values of £10, £14, £32 and £34 on the class 19 imply that r stabilizes these 4 characters. It follows that 8 if d = 11 and w = 2, 6 if d = 10 and w = 2, 4 if d = 10 and w = 1, 3 if d = 9 and w = 2, k (N{C(3)),B0,d,w) = (3.7) 2 if d = 9 and w = 1, 6 if d = 8 and w = 2 , 6 if d = 7 and w = 2, 0 otherwise. (3) Suppose C = C(4), so that N(C) ~ E± . [28] : S3 and its ordinary character table is given by Table 7. The calculation of the numbers k(G, B0 ,d, w) for C is also similar to that for C(2) above. First of all, r stabilizes the following characters of Irr(6 0): {& : * e {1,2,4,8,9,12,14,16,17,20,27}}; DADE’S CONJECTURE FOR THE TITS GROUP 117 and r permutes each of the following pairs of characters: (£3, £5), (£6, £11), (£7, £10), (£13,6 5 ), (£18,£19), (£21,£22), (£23,£26) and (£24,£25)- By Table 10, £ [8 = £19, ££x = £22, £[3 = £15 and r stabilizes the other characters except possibly the pair (£23,£26), in which case (xk(C ),£23> = (xIat(C),£26) for all x € Irr(G). However, the values of £17, £24 and £25 on the classes 12 and 13 imply that these two classes are stabilized by r, and this implies that £23 and £26 are stabilized by r. It follows that 8 if d = 11 and w = 2, 2 if d = 10 and w = 2, 4 if d = 10 and w = 1, 3 if d = 9 and w = 2, 2 if d = 9 and w = 1, (3.8) 4 if d — 8 and w = 2, 2 if d = 7 and w = 2, 2 if d = 7 and w = 1, v0 otherwise. By (3A) (c), (3.6), (3.7) and (3.8), 4 E (-l)‘ k(JV(C(i)),B0,d,w ) = 0 (3.9) 2=1 for all integers d,w > 0. Thus (3C) follows. □ (3 D ). Let B be a p-block of the simple Tits group G. Then the number W (B ) of B-weights is equal to the number i(B ) of irreducible Brauer characters in B . P roof. If B has a cyclic defect group, then the Alperin weight conjecture follows by [7, Theorem 9.1] and [12, Proposition 5.6]. We may suppose B has a non-cyclic defect group, so that p € {2,3,5} and B = B q. By [4, (1.3)], W (B 0) = E|Irr°(iV(fl)/C(iJ)i?)|, R where R runs over non-trivial subgroups in $ (G ,p ) with a(C(R )R ) = a(R). If p = 5, then R = £25 , N(R)/C(R)R — Z4 . A*, and W(i?o) = k(^4 • A 4 ) = 14. Thus W(B0) = t(B0) by (3A) (d). If p = 3, then R e {Eg^^S1^2}. Thus N{C)/C(R)R ~ GL2(3) or SU3(2)/ 3+ "2 ~ Qs according as \R\ = 9 or \R\ = 27, where Q% is a quaternion group. Now GL/2(3 ) has exactly 2 irreducible characters (the Steinberg characters) of 3-defect 0 , and Irr(Qg) has exactly 5 irreducible characters of 3-defect 0 . So W {B0) = 9 = £(B0). If p = 2, then by (2.5), R € {2 . [28], 22 . [28], 5 }, where S is a Sylow 2-subgroup of G. Since each Irr° (N(R)/C(R)R) contains exactly one character, it follows that W (B Q) = 3 = £{B0). This proves (3D). □ 118 JIANBEI AN Acknow ledgem ents. The author would like to thank Professor Dade for useful suggestions and corrections of (2A) (c). The author also would like to thank Pro fessors Marston Conder and Peter Lorimer for much useful help, and the Auckland University Research Committee for its support. References 1. J.L. Alperin, Weights for finite groups , in The Areata Conference on Represen tations of Finite Groups, Proc. of Symposia in Pure Math. 47 (1987), 369-379. 2. Jianbei An, Dade’s conjecture for the Ree groups 2 F2 (q2) in non-defining char acteristics, submitted. 3. Jianbei An, Dade’s conjecture for the Chevalley groups G 2 (q) in non-defining characteristics, Canadian J. Math. 48 (1996), 671-691. 4. Jianbei An and M. Conder, The Alperin and Dade conjectures for the simple Mathieu groups, Comm. Algebra 23 (1995), 2797-2823. 5. J.J. Cannon, An Introduction to the Group Theory Language C A Y L E Y , Com putational Group Theory, Academic Press, New York, 1984. 6. J.H. Conway, R.T. Curtis, S.P. Parker and R.A. Wilson, An ATLAS of Finite Groups , University Press, Oxford, 1985. 7. E. Dade, Counting characters in blocks, I, Invent. Math. 109 (1992), 187-210. 8. E. Dade, Counting characters in blocks, II , J. reine angew. Math., 448 (1994), 97-190. 9. E. Dade, Counting characters in blocks , 2.9, preprint. 10. E. Dade, Private communication. 11. W. Feit, The Representation Theory of Finite Groups , North Holland, 1982. 12. R. Knorr and G. Robinson, Some remarks on a conjecture of Alperin, J. London Math. Soc. 39 (1989), 48-60. 13. G. Malle, The maximal subgroups of 2 F±(q2), J. Algebra 139 (1989), 52-69. 14. G. Malle, Die unipotenten charaktere von 2 F2 (q2), Comm. Algebra 18 (1990), 2361-2381. 15. R.A. Wilson, The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits, Proc. London Math. Soc. 48 (1984), 533-563. A ppendix Let G be the simple Tits group and C (j) a radical p-chains defined by (2.2), (2.4) and (2.5). The structure of the normalizers N ( C ( j)) = A^c((7(j)) in G is given by either (2A) (b), (c) or (2B) (b), and its character table can be obtained by CAYLEY. In Tables 1-3, we give the degrees of irreducible characters of N G (C (j)), where each C (j) is given by (2.4) with j > 2. In Tables 4-7, we give the character tables of ordinary irreducible characters of N c (C (j)) of C (j) with j > 2. In Tables 8-10 we give the decompositions of the induced character IndG ^ ^ (£) of £ G Irr (N (C (j))) as a linear combination of irreducible characters of G. In Tables 4-7, i — v^T , j = e ^ , a = 1 + j , b = 1 — j , c = — 1 — 2j, d = — 1 -I- 2j, w = 2 t t 5 x = V3, y — y/ 2 and 2 = —w3 — w7. DADE’S CONJECTURE FOR THE TITS GROUP 119 T a b le 1. The degrees of characters of Irr(3++ 2 :Dg) character £1 £2 £3 £4 £5 £6 £7 £8 £9 £10 6 1 £12 £13 degree 1 1112 4444 6 6 6 6 T a b le 2. The degrees of characters of Irr(3++2 : £4 ) character £1 £2 £3 £4 £5 £6 £7 £8 £9 £10 £11 order 1 1 1 1 2 2 2 2 4 6 6 T a b le 3. The degrees of characters of Irr(Eg : GL2(3)) character £1 £2 £3 £4 £5 £6 £7 £8 £9 £10 £11 order 1 1 2 2 2 3 3 4 8 8 16 T a b le 4. The ordinary character table of the groupE 25 : Z 4.A4 class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 order 1 2 2 3 3 4 4 4 5 6 6 10 12 12 12 12 £1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 £2 1 1 1 -a j 1 1 1 1 -a j 1 j -a j -a £3 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 £4 1 1 -1 -a j -1 -1 1 1 -a j -1 -j a -j a £5 1 1 1 j -a 1 1 1 1 j -a 1 -a j -a j & 1 1 -1 j -a -1 -1 1 1 j -a -1 a -j a -j £7 2 -2 0 -j a 2i -2i 0 2 j -a 0 z w7 -z -w7 £8 2 -2 0 -1 -1 2i -2i 0 2 1 1 0 i i -i -i £9 2 -2 0 -1 -1 -2i 2i 0 2 1 1 0 -i -i i i 7 7 £10 2 -2 0 -j a -2i 2i 0 2 j -a 0 -z -w z w 7 7 £11 2 -2 0 a -j -2i 2i 0 2 -a j 0 -w -z w z £12 2 -2 0 a -j 2i -2i 0 2 -a j 0 w7 z -w7 -z £13 3 3 1 0 0 -3 -3 -1 3 0 0 1 0 0 0 0 £14 3 3 -1 0 0 3 3 -1 3 0 0 -1 0 0 0 0 £15 24 0 -4 0 0 0 0 0 -1 0 0 1 0 0 0 0 £l6 24 0 4 0 0 0 0 0 -1 0 0 -1 0 0 0 0 1 2 0 JIANBEI AN T a b le 5. The ordinary character table of Z2 . [28] : E5 : Z 4 class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 conjg 1 1 10 20 80 80 80 80 160 160 160 320 320 320 order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 $1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 $2 1 1 1 1 1 -1 -1 1 1 -1 1 -i i -i $3 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 $4 1 1 1 1 1 -1 -1 1 1 -1 1 i -i i $5 4 4 4 4 4 4 0 0 0 $6 5 5 5 5 -3 1 1 -3 1 1 1 1 1 1 $7 5 5 5 5 -3 1 1 -3 1 1 1 -1 -1 -1 $8 5 5 5 5 -3 -1 -1 -3 1 -1 1 i -i i $9 5 5 5 5 -3 -1 -1 -3 1 -1 1 -i i -i $10 10 10 -6 2 2 2 2 -2 2 2 -2 0 0 0 $11 10 10 -6 2 2 0 0 -2 -2 0 2 -b -a -b $12 10 10 -6 2 2 -2 -2 -2 2 -2 -2 0 0 0 $13 10 10 -6 2 2 0 0 -2 -2 0 2 -a -b -a $14 10 10 -6 2 2 -2 -2 -2 2 -2 -2 0 0 0 $15 10 10 -6 2 2 0 0 -2 -2 0 2 a b a $16 10 10 -6 2 2 0 0 -2 -2 0 2 b a b $17 10 10 -6 2 2 2 2 -2 2 2 -2 0 0 0 $18 10 10 10 10 2 2 2 2 -2 2 -2 0 0 0 $19 10 10 10 10 2 -2 -2 2 -2 -2 -2 0 0 0 $20 16 -16 0 0 0 -4 4 0 0 0 0 -2i 2i 2i $21 16 -16 0 0 0 -4 4 0 0 0 0 2i -2i -2i $22 16 -16 0 0 0 4 -4 0 0 0 0 -2 -2 2 $23 16 -16 0 0 0 4 -4 0 0 0 0 2 2 -2 $24 20 20 -12 4 -4 0 0 4 0 0 0 0 0 0 $25 20 20 -12 4 -4 0 0 4 0 0 0 0 0 0 $26 40 40 8 -8 0 4 4 0 0 -4 0 0 0 0 $27 40 40 8 -8 0 -4 -4 0 0 4 0 0 0 0 $28 64 -64 0 0 0 0 0 0 0 0 0 0 0 0 DADE’S CONJECTURE FOR THE TITS GROUP 1 2 1 T able 5 continued The ordinary character table of . [28]:E 5 : Z 4 class 15 16 17 18 19 20 21 22 23 24 25 26 27 28 conjg 320 320 1024 320 320 640 640 640 640 1024 640 640 640 640 order 4 4 5 8 8 8 8 8 8 10 16 16 16 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 £2 i -1 1 -1 -1 -1 -1 i -i 1 -i i -i i £3 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 £4 -i -1 1 -1 -1 -1 -1 -i i 1 i -i i -i £5 0 -1 0 0 -1 0 0 0 0 $6 1 1 0 1 1 -1 -1 1 1 0 -1 -1 -1 -1 £7 -1 1 0 1 1 -1 -1 -1 -1 0 1 1 1 1 & -i -1 0 -1 -1 1 1 -i i 0 -i i -i i £9 i -1 0 -1 -1 1 1 i -i 0 i -i i -i £l0 0 -2 0 0 0 0 0 0 0 0 y -y -y y -a 0 0 -2i 2i 0 0 a b 0 0 0 0 0 £l2 0 2 0 0 0 0 0 0 0 0 yi yi -yi -yi £l3 -b 0 0 2i -2i 0 0 b a 0 0 0 0 0 £l4 0 2 0 0 0 0 0 0 0 0 -yi -yi yi yi £l5 b 0 0 2i -2i 0 0 -b -a 0 0 0 0 0 £l6 a 0 0 -2i 2i 0 0 -a -b 0 0 0 0 0 £l7 0 -2 0 0 0 0 0 0 0 0 -y y y -y £l8 0 2 0 -2 -2 0 0 0 0 0 0 0 0 0 £l9 0 -2 0 2 2 0 0 0 0 0 0 0 0 0 £20 -2i 0 1 0 0 0 0 0 0 -1 0 0 0 0 £21 2i 0 1 0 0 0 0 0 0 -1 0 0 0 0 £22 2 0 1 0 0 0 0 0 0 -1 0 0 0 0 £23 -2 0 1 0 0 0 0 0 0 -1 0 0 0 0 £24 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 £25 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 £26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £28 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 1 2 2 JIANBEI AN T a b le 6 . The ordinary character table of Z2 . [28] : Z 4 class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 conjg 1 1 2 4 8 8 8 16 16 16 64 16 32 32 32 64 64 64 order 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 €l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 £2 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 1 1 €3 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 €4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 €5 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 1 1 1 €6 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 €7 1 1 1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 $8 1 1 1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 €9 2 2 2 2 -2 2 -2 -2 -2 2 0 -2 -2 2 -2 0 0 0 €l0 2 2 2 2 -2 2 -2 0 0 2 0 -2 2 -2 0 0 0 0 € ll 2 2 2 2 -2 2 -2 0 0 2 0 -2 2 -2 0 0 0 0 £l2 2 2 2 2 2 2 2 -2 -2 2 0 2 2 2 -2 0 -2 0 £l3 2 2 2 2 -2 2 -2 2 2 2 0 -2 -2 2 2 0 fl 0 €l4 2 2 2 2 -2 2 -2 0 0 2 0 -2 2 -2 0 0 0 0 $15 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 0 -2 0 Cl6 2 2 2 2 -2 2 -2 0 0 2 0 -2 2 -2 0 0 0 0 Cl7 2 2 2 2 -2 2 -2 -2 -2 2 0 -2 -2 2 -2 0 0 0 €18 2 2 2 2 -2 2 -2 2 2 2 0 -2 -2 2 2 0 0 0 €19 4 4 4 4 -4 4 -4 0 0 -4 0 4 0 0 0 0 0 0 €20 4 4 4 4 -4 4 -4 0 0 -4 0 4 0 0 0 0 0 0 €21 4 4 4 4 4 4 4 0 0 4 0 4 -4 -4 0 0 0 0 €22 4 4 4 4 4 4 4 0 0 -4 -2 -4 0 0 0 -2 0 2 £23 4 4 4 4 4 4 4 0 0 -4 2 -4 0 0 0 2 0 -2 €24 8 8 -8 0 4 0 -4 0 0 0 -2 0 0 0 0 2 2 2 $25 8 8 8 8 0 -8 0 4 4 0 0 0 0 0 -4 0 0 0 £26 8 8 8 8 0 -8 0 -4 -4 0 0 0 0 0 4 0 0 0 ?27 8 8 -8 0 4 0 -4 0 0 0 -2 0 0 0 0 2 -2 -2 $28 8 8 -8 0 4 0 -4 0 0 0 2 0 0 0 0 -2 2 -2 ^29 8 8 -8 0 4 0 -4 0 0 0 2 0 0 0 0 -2 -2 2 €30 16 -16 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 ^31 16 16 -16 0 -8 0 8 0 0 0 0 0 0 0 0 0 0 0 $32 16 -16 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 £33 16 -16 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 €34 16 -16 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 £35 16 16 16 -16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 DADE’S CONJECTURE FOR THE TITS GROUP 123 T a b le 6 continued. The ordinary character table of Z2 . [28] : Z4 class 19 20 21 22 23 24 25 26 27 28 29 30 31 32 3 3 34 35 conjg 64 64 64 64 64 64 64 64 64 128 128 128 128 128 128 128 128 order 4 4 4 4 4 4 4 8 8 8 8 8 8 16 16 16 16 €1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 £2 £3 1 1 1 1 1 1 _1 1 1 1 1 -1 -1 -1 -1 -1 -1 £4 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 £5 -i i £6 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 £7 i -i i -i -1 1 -1 -1 -1 i -i 1 1 -i i -i i £8 -i i -i i -1 1 -1 -1 -1 -i i 1 1 i -i i -1 £9 0 0 0 0 2 0 0 0 0 0 0 0 0 yi yi -yi -yi £10 -b -a -b -a 0 0 0 -2i 2i b a 0 0 0 0 0 0 £11 b a b a 0 0 0 -2i 2i -b -a 0 0 0 0 0 0 £12 0 0 0 0 -2 -2 0 2 2 0 0 0 0 0 0 0 0 £13 0 0 0 0 -2 0 0 0 0 0 0 0 0 y -y -y y £14 -a -b -a -b 0 0 0 2i -2i a b 0 0 0 0 0 0 €15 0 0 0 0 2 -2 0 -2 -2 0 0 0 0 0 0 0 0 £16 a b a b 0 0 0 2i -2i -a -b 0 0 0 0 0 0 £17 0 0 0 0 2 0 0 0 0 0 0 0 0 -yi -yi y y £l8 0 0 0 0 -2 0 0 0 0 0 0 0 0 -y y y -y £19 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 €20 0 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 £21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £22 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 £23 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 £24 0 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 0 0 £25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £27 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 £28 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 £29 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 0 0 £30 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 £31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 £32 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 0 0 £33 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 £34 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 0 0 £35 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 124 JIANBEI AN T a b le 7. The ordinary character table of i?4[28] . S3 class 1 2 3 4 5 6 7 8 9 10 11 12 13 14 conjg 1 3 4 24 24 24 192 512 32 48 96 192 192 192 order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 £2 1 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 £3 2 2 2 2 -2 -2 0 -1 -2 -2 2 0 0 0 £4 2 2 2 2 2 2 0 -1 2 2 2 0 2 0 £5 2 2 2 2 -2 -2 0 -1 -2 -2 2 0 0 0 & 3 3 3 3 -1 -1 1 0 3 -1 -1 1 1 1 £7 3 3 3 3 -1 -1 -1 0 3 -1 -1 -1 1 -1 3 3 3 3 3 3 -1 0 3 3 3 -1 -1 -1 £9 3 3 3 3 3 3 1 0 3 3 3 1 -1 1 £10 3 3 3 3 -1 -1 -1 0 3 -1 -1 -1 1 -1 61 3 3 3 3 -1 -1 1 0 3 -1 -1 1 1 1 £12 4 4 4 4 -4 -4 0 1 -4 -4 4 0 0 0 £13 6 6 6 6 2 2 0 0 -6 2 -2 0 0 0 £14 6 6 6 6 -2 -2 0 0 6 -2 -2 0 -2 0 £15 6 6 6 6 2 2 0 0 -6 2 -2 0 0 0 £l6 12 12 12 -4 4 4 -2 0 0 -4 0 2 0 2 £l7 12 12 12 -4 4 4 2 0 0 -4 0 -2 0 -2 £l8 12 12 12 -4 -4 -4 0 0 0 4 0 0 0 0 £l9 12 12 12 -4 -4 -4 0 0 0 4 0 0 0 0 £20 16 16 -16 0 0 0 0 -2 0 0 0 0 0 0 £21 16 16 -16 0 0 0 0 1 0 0 0 0 0 0 £22 16 16 -16 0 0 0 0 1 0 0 0 0 0 0 £23 24 -8 0 0 4 -4 -2 0 0 0 0 -2 2 2 £24 24 -8 0 0 4 -4 2 0 0 0 0 -2 -2 2 £25 24 -8 0 0 4 -4 2 0 0 0 0 2 2 -2 £26 24 -8 0 0 4 -4 -2 0 0 0 0 2 -2 -2 £27 48 -16 0 0 -8 8 0 0 0 0 0 0 0 0 DADE’S CONJECTURE FOR THE TITS GROUP 125 T a b le 7 continued. The ordinary character table of £ 4 [28] . S3 class 15 16 17 18 19 20 21 22 23 24 25 26 27 conjg 192 192 512 192 192 384 384 512 512 384 384 384 384 order 4 4 6 8 8 8 8 12 12 16 16 16 16 1 1 1 1 1 1 1 1 1 1 1 1 1 £2 -1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 £3 0 0 -1 0 0 0 0 1 1 yi yi -yi -yi £4 0 2 -1 2 2 0 0 -1 -1 0 0 0 0 £5 0 0 -1 0 0 0 0 1 1 -yi -yi yi yi £e 1 1 0 c d -1 -1 0 0 i -i i -i £7 -1 1 0 c d 1 1 0 0 -i i -i i £s -1 -1 0 -1 -1 -1 -1 0 0 1 1 1 1 £9 1 -1 0 -1 -1 1 1 0 0 -1 -1 -1 -1 £10 -1 1 0 d c 1 1 0 0 i -i i -i £11 1 1 0 d c -1 -1 0 0 -i i -i i £12 0 0 1 0 0 0 0 -1 -1 0 0 0 0 £13 0 0 0 0 0 0 0 0 0 y -y -y y £14 0 -2 0 2 2 0 0 0 0 0 0 0 0 £15 0 0 0 0 0 0 0 0 0 -y y y -y £l6 -2 0 0 0 0 0 0 0 0 0 0 0 0 Cl7 2 0 0 0 0 0 0 0 0 0 0 0 0 £l8 0 0 0 0 0 -2 2 0 0 0 0 0 0 £l9 0 0 0 0 0 2 -2 0 0 0 0 0 0 £20 0 0 2 0 0 0 0 0 0 0 0 0 0 £21 0 0 -1 0 0 0 0 X -X 0 0 0 0 £22 0 0 -1 0 0 0 0 -x x 0 0 0 0 £23 2 -2 0 0 0 0 0 0 0 0 0 0 0 £24 -2 2 0 0 0 0 0 0 0 0 0 0 0 £25 -2 -2 0 0 0 0 0 0 0 0 0 0 0 £26 2 2 0 0 0 0 0 0 0 0 0 0 0 £27 0 0 0 0 0 0 0 0 0 0 0 0 0 126 JIANBEI AN T able 8. The decomposition of IndG2 [28] s3 (&) Xi X2 X3 X4 X5 X6 X7 X8 X9 X10 X 11 6 1 0 0 0 0 1 0 0 0 1 0 $2 0 0 0 1 0 0 0 0 0 0 0 $3 0 0 0 0 0 0 0 0 0 1 0 & 0 0 0 0 1 0 0 0 0 0 0 £5 0 0 0 0 0 0 0 0 0 1 0 & 0 0 0 0 0 1 0 1 0 1 0 & 0 0 0 0 0 0 0 0 0 0 0 & 0 0 0 0 0 0 0 0 0 0 1 £9 0 0 0 0 0 0 0 0 1 0 0 £l0 0 0 0 0 0 0 0 0 0 1 0 £ll 0 0 0 0 0 0 0 0 0 0 0 £l2 0 0 1 0 0 0 1 0 0 0 0 ^13 0 0 0 0 0 0 0 0 0 0 0 Cl4 0 1 0 0 0 0 1 0 0 0 0 ^15 0 0 0 1 0 0 0 0 1 1 0 £l6 0 0 0 0 1 0 0 0 0 1 1 £l7 0 0 0 0 0 0 0 0 0 1 0 £l8 0 0 0 0 0 0 0 0 0 2 0 £l9 0 0 0 0 0 0 0 0 0 0 0 £20 0 0 0 0 0 1 0 1 1 1 1 £21 0 0 0 0 0 1 0 1 1 1 1 £22 0 1 1 0 0 0 2 0 0 0 0 £23 0 0 0 1 1 0 0 0 1 0 1 £24 0 0 0 0 0 0 1 1 1 0 1 £25 0 0 0 0 0 0 1 1 1 0 1 £26 0 0 0 0 0 1 0 1 1 3 1 £27 0 0 0 0 0 0 2 2 2 0 2 Uy> 00 0 0 0 0 0 0 2 2 2 2 2 DADE’S CONJECTURE FOR THE TITS GROUP 127 T a b l e 8 continued. The decomposition of IndG2 r28i s3(£t) X12 Xl3 Xl4 Xl5 Xie K17 Xi8 Xl9 X20 X21 X22 6 0 0 1 1 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 1 0 0 £3 0 0 0 0 1 1 0 0 0 0 0 £4 0 0 0 0 0 0 0 0 1 0 0 £5 1 1 1 1 0 0 0 0 0 1 1 £6 0 0 1 1 0 0 1 1 0 1 1 £7 0 0 0 1 1 1 1 1 0 1 1 & 0 0 0 0 0 0 1 1 1 1 1 £9 0 0 0 0 0 0 1 1 1 1 1 £10 1 1 1 1 1 2 1 1 1 2 2 £11 1 1 1 0 1 1 1 1 2 2 2 £12 1 1 0 0 0 0 1 1 3 2 2 £13 1 1 0 1 1 1 1 2 2 2 £14 1 1 0 0 0 0 1 1 3 2 2 £15 1 1 1 1 0 0 1 1 2 2 2 £l6 1 1 1 1 0 0 1 1 2 2 2 £l7 1 1 1 1 2 1 1 1 1 2 2 UJ> 00 1 1 1 2 2 2 1 1 0 2 2 £19 1 1 1 0 1 1 1 1 2 2 2 £20 0 0 1 2 2 2 1 3 3 £21 0 0 1 2 2 2 1 3 3 £22 2 2 0 0 1 1 1 1 5 3 3 £23 2 2 2 0 0 0 1 1 5 3 3 £24 1 1 1 1 1 1 2 3 4 4 4 £25 1 1 1 1 1 1 3 2 4 4 4 £26 2 2 3 4 4 4 6 6 4 8 8 £27 2 2 2 2 2 2 5 5 8 8 8 £28 4 4 4 4 4 4 8 8 11 13 13 128 JIANBEI AN T able 9. The decomposition of IndGj28j 4(£i) Xi X2 X3 X4 X5 X6 X7 X8 X9 X10 Xu 6 1 0 0 0 0 1 0 0 0 2 0 6 0 0 0 0 1 0 0 0 0 1 0 £3 0 0 0 0 0 1 0 1 0 1 0 £4 0 0 0 0 0 0 0 0 0 2 0 & 0 0 0 1 0 0 0 0 0 1 0 £6 0 0 0 0 0 0 0 0 0 0 0 & 0 0 0 0 0 0 0 0 0 0 1 & 0 0 0 0 0 0 0 0 1 0 0 £9 0 1 0 0 0 0 1 0 0 0 0 £10 0 0 0 0 0 0 0 0 0 0 0 £11 0 0 0 0 1 0 0 0 0 1 1 £12 0 0 0 0 0 0 0 0 0 0 0 £13 0 0 0 0 0 0 0 0 0 1 0 £14 0 0 0 0 0 0 0 0 0 0 0 £15 0 0 0 0 0 0 0 0 0 2 0 £l6 0 0 0 1 0 0 0 0 1 1 0 £l7 0 0 1 0 0 0 1 0 0 0 0 £l8 0 0 0 0 0 0 0 0 0 1 0 £l9 0 0 0 0 0 0 1 1 1 0 1 £20 0 0 0 0 0 0 1 1 1 0 1 £21 0 0 0 0 0 0 0 0 0 2 0 £22 0 0 0 0 0 1 0 1 1 1 1 £23 0 0 0 0 0 0 0 0 0 2 0 £24 0 0 0 0 0 0 2 2 2 0 2 £25 0 0 0 0 0 1 0 1 1 3 1 £26 0 0 0 0 0 0 2 2 2 0 2 £27 0 0 0 0 0 0 2 2 2 0 2 £28 0 1 1 0 0 0 2 0 0 2 0 £29 0 0 0 1 1 0 0 0 1 2 1 £30 0 1 1 0 0 0 4 2 2 2 2 £31 0 0 0 0 0 1 2 3 3 3 3 £32 0 0 0 0 0 1 2 3 3 3 3 £33 0 0 0 1 1 0 2 2 3 2 3 £34 0 0 0 0 0 1 2 3 3 3 3 £35 0 0 0 0 0 1 2 3 3 3 3 DADE’S CONJECTURE FOR THE TITS GROUP 129 T able 9 continued. The decomposition of IndGj28j 4(£i) Xl2 Xl3 X14 Xl5 Xi6 Xl7 Xi8 Xl9 X20 X21 X22 £l 1 1 2 2 0 0 0 0 0 1 1 £2 1 1 1 1 0 0 0 0 1 1 1 £3 0 0 1 1 0 0 1 1 0 1 1 £4 1 1 1 1 1 1 0 0 0 1 1 £5 1 1 1 1 0 0 0 0 1 1 1 & 0 0 0 1 1 1 1 1 0 1 1 £7 0 0 0 0 0 0 1 1 1 1 1 £s 0 0 0 0 0 0 1 1 1 1 1 £9 1 1 0 0 0 0 1 1 3 2 2 £10 1 1 1 0 1 1 1 1 2 2 2 £11 1 1 1 1 0 0 1 1 2 2 2 £12 1 1 1 0 1 1 1 1 2 2 2 £13 1 1 1 1 1 2 1 1 1 2 2 £14 1 1 1 0 1 1 1 1 2 2 2 £15 1 1 1 2 2 2 1 1 0 2 2 £l6 1 1 1 1 0 0 1 1 2 2 2 £l7 1 1 0 0 0 1 1 3 2 2 £l8 1 1 1 1 2 1 1 1 1 2 2 £l9 1 1 1 1 1 1 2 3 4 4 4 ^20 1 1 1 1 1 1 3 2 4 4 4 ^21 2 2 2 2 3 3 2 2 2 4 4 £22 0 0 1 2 1 1 4 4 2 4 4 £23 2 2 2 2 3 3 2 2 2 4 4 £24 2 2 2 2 2 2 5 5 8 8 8 £25 2 2 3 4 4 4 6 6 4 8 8 £26 2 2 2 2 2 2 5 5 8 8 8 £27 2 2 2 2 2 2 5 5 8 8 8 £28 4 4 2 2 3 3 4 4 8 8 8 £29 4 4 4 2 2 2 4 4 8 8 8 £30 6 6 4 4 5 5 9 9 16 16 16 £31 4 4 5 6 6 6 11 11 12 16 16 £32 4 4 5 6 6 6 11 11 12 16 16 £33 6 6 6 4 4 4 9 9 16 16 16 £34 4 4 5 6 6 6 11 11 12 16 16 £35 4 4 5 6 6 6 11 11 12 16 16 130 JIANBEI AN T able 10. The decomposition of In d ^ 28].5.4(£i) Xi X2 X3 X4 X5 X6 X7 X8 X9 X10 Xi: 6 1 0 0 0 0 0 0 0 0 1 0 £2 0 0 0 0 0 0 0 1 0 0 0 £3 0 0 1 0 0 0 0 0 0 0 0 £4 0 0 0 0 0 1 0 0 0 1 0 £5 0 1 0 0 0 0 0 0 0 0 0 £6 0 0 0 1 0 0 0 0 0 1 0 £7 0 0 0 0 0 0 0 0 1 0 0 £8 0 0 0 0 0 0 0 0 0 0 0 £9 0 0 0 0 0 0 0 0 0 2 0 £10 0 0 0 0 0 0 0 0 0 0 1 6 1 0 0 0 0 1 0 0 0 0 1 0 £12 0 0 0 0 0 0 1 0 0 0 0 £13 0 0 0 0 0 0 0 0 0 1 0 £14 0 0 0 0 0 0 0 0 0 0 0 £15 0 0 0 0 0 0 0 0 0 1 0 £16 0 0 0 0 0 1 0 1 1 1 1 £17 0 0 0 0 0 0 0 0 0 2 0 £l8 0 0 0 0 0 0 1 1 1 0 1 Cl9 0 0 0 0 0 0 1 1 1 0 1 £20 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 £22 0 0 0 0 0 0 1 1 1 1 1 £23 0 0 0 0 0 0 2 2 2 0 2 £24 0 1 1 0 0 0 2 0 0 2 0 £25 0 0 0 1 1 0 0 0 1 2 1 £26 0 0 0 0 0 0 2 2 2 0 2 £27 0 0 0 0 0 1 2 3 3 3 3 DADE’S CONJECTURE FOR THE TITS GROUP T able 10 continued. The decomposition of IndGj28].5.4(£i) X.12 Xl3 Xl4 Xl5 Xi6 Xl7 Xi8 Xl9 X20 X21 X22 £1 1 1 1 1 0 0 0 0 0 0 0 £2 0 0 0 0 0 0 1 1 0 0 0 £3 0 0 0 0 0 0 0 0 1 1 1 £4 0 0 1 1 0 0 0 0 0 1 1 £5 0 0 0 0 0 0 0 0 1 1 1 £6 1 1 1 1 0 0 0 0 1 1 1 £7 0 0 0 0 0 0 1 1 1 1 1 £8 0 0 0 1 1 1 1 1 0 1 1 £9 1 1 1 1 1 1 0 0 0 1 1 £10 0 0 0 0 0 0 1 1 1 1 1 £11 1 1 1 1 0 0 0 0 1 1 1 £12 1 1 0 0 0 0 1 1 2 1 1 £13 1 1 1 1 1 2 1 1 1 2 2 £14 1 1 1 0 1 1 1 1 2 2 2 £15 1 1 1 1 2 1 1 1 1 2 2 £16 0 0 1 2 1 1 4 4 2 4 4 £17 2 2 2 2 3 3 2 2 2 4 4 £l8 1 1 1 1 1 1 3 2 4 4 4 £l9 1 1 1 1 1 1 2 3 4 4 4 £20 1 1 1 2 2 2 3 3 4 6 6 £21 1 2 2 2 2 2 4 4 4 5 5 £22 2 1 2 2 2 2 4 4 4 5 5 £23 2 2 2 2 2 2 5 5 8 8 8 £24 4 4 2 2 3 3 4 4 8 8 8 £25 4 4 4 2 2 2 4 4 8 8 8 £26 2 2 2 2 2 2 5 5 8 8 8 £27 4 4 5 6 6 6 11 11 12 16 16 Jianbei An The University of Auckland Private Bag 92019 Auckland NEW ZEALAND an@mat h. auckland. ac. nz