(3, 𝑞, 𝑟)-generations of Fischer's sporadic group Fi′24

Faryad Ali; Mohammed Ali Faya Ibrahim; Andrew Woldar

doi:10.1515/jgth-2018-0104

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(3, 𝑞, 𝑟)-generations of Fischer's sporadic group Fi′₂₄

Faryad Ali , Mohammed Ali Faya Ibrahim und Andrew Woldar

Veröffentlicht/Copyright: 20. Januar 2019

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Aus der Zeitschrift Journal of Group Theory Band 22 Heft 3

Abstract

A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o⁢(x)=l, o⁢(y)=m and o⁢(x⁢y)=n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer’s largest sporadic simple group Fi24′ by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi24′| with 3<q<r.

1 Introduction

A group G is said to be (l,m,n)-generated if G=〈x,y〉, where the elements x,y,x⁢y have respective orders o⁢(x)=l, o⁢(y)=m, o⁢(x⁢y)=n. In such case, G is a quotient group of the Von Dyck group D⁢(l,m,n), and therefore it is also (π⁢(l),π⁢(m),π⁢(n))-generated for any π∈S3. Thus one may assume straightaway that l≤m≤n.

Initially, the study of (l,m,n)-generations of a group G had deep connections to the topological problem of determining the least genus of an orientable surface on which G admits an effective, orientation-preserving, conformal action. In [27], J. Moori extended these investigations beyond the “minimum genus problem” to all possible (p,q,r)-generations, assuming G to be a finite non-abelian simple group and p,q,r to be distinct primes. Generational results of this type have since proved to be quite useful and interesting.

Note that if p divides l and q divides m, then (p,q,r)-generation implies (l,m,n)-generation for some value of n; however, the precise determination of n requires additional work, which oftentimes is quite substantial.

Groups that can be generated by an involution and an element of order 3 are said to (2,3)-generated, and such generations have been of particular interest to combinatorists and group theorists. The quintessential example of a (2,3)-generated group is the modular group PSL⁢(2,ℤ), which, being the free product of the groups ℤ2 and ℤ3, acts as a universal cover. This implies that any group generated by an involution and an element of order 3 is a quotient group of PSL⁢(2,ℤ). Connections with Hurwitz groups, regular maps, Beauville surfaces and structures provide additional motivation for the study of these groups. (Recall that a Hurwitz group is one that can be (2,3,7)-generated.)

If a simple group G is (l,m,n)-generated, then by Conder [9] either G≅A5 or 1l+1m+1n<1. In particular, this implies that every simple (3,m,n)-generated group acts conformally on a closed compact Riemann surface obtained as the quotient of a discrete holomorphic action of D⁢(3,m,n) on the hyperbolic plane ℋ.

In [21, 20, 19, 22, 23, 27], Ganief and Moori established all possible (p,q,r)-generations for the sporadic groups He, HS, Co2, Co3, J1, J2, J3, J4, Fi22, where p,q,r are distinct prime divisors of the group order. Further, Darafsheh, Ashrafi and their co-authors determined such (p,q,r)-generations for the sporadic groups He, Co1, Ly, O’N, HN, Ru, Th (cf. [13, 12, 15, 14, 16, 6, 7, 8]). More recently (cf. [3, 5]), the authors determined all (2,q,r)-generations for Fischer’s sporadic simple groups Fi23 and Fi24′. (See also [1, 2, 17, 24] for further details on the generation of finite simple groups by two elements.)

The focus of the present article is on (3,q,r)-generations of Fi24′, where q and r are prime divisors of |Fi24′| with 3<q<r. The case of (p,q,r)-generation of Fi24′ where 5≤p<q<r will be treated in a forthcoming article.

For Fi24′-conjugacy classes 3⁢X, qY, rZ, let us refer to Fi24′ as (3⁢X,q⁢Y,r⁢Z)-generated provided Fi24′=〈x,y〉, where x∈3⁢X, y∈q⁢Y, x⁢y∈r⁢Z. We may now state the main result of our paper.

Theorem.

Let q and r be prime divisors of |Fi24′| with 3<q<r. Then Fischer’s largest sporadic simple group Fi24′ is (3⁢X,q⁢Y,r⁢Z)-generated if and only if the triple (3⁢X,q⁢Y,r⁢Z) differs from (3⁢A,5⁢A,7⁢A),(3⁢A,5⁢A,7⁢B),(3⁢A,5⁢A,13⁢A).

2 Preliminaries

Throughout this article, we use the same notation and terminology as can be found in [1, 2, 3, 16, 22, 27, 28, 30]. In particular, for a finite group G with conjugacy classes C1, C2, C3, we denote by ΔG⁢(C1,C2,C3) the corresponding structure constant of G. Observe that ΔG⁢(C1,C2,C3) is nothing more than the cardinality of the set Ω={(x,y)∣x⁢y=z}, where x∈C1, y∈C2 and z is a fixed representative in the conjugacy class C3. It is well known that the value of ΔG⁢(C1,C2,C3) can be computed from the character table of G (e.g., see [25, p.45]) via the formula

ΔG⁢(C1,C2,C3)=|C1|⁢|C2||G|⁢∑i=1mχi⁢(x)⁢χi⁢(y)⁢χi⁢(z)¯[χi⁢(1G)],

where χ1,χ2,…,χm are the irreducible complex characters of G, and the bar denotes complex conjugation. We denote by ΔG*⁢(C1,C2,C3) the number of distinct ordered pairs (x,y)∈Ω such that G=〈x,y〉. Clearly, if ΔG*⁢(C1,C2,C3)>0, then G is (l,m,n)-generated, where l,m,n are the respective orders of elements from C1,C2,C3. In this instance, we say that G is (C1,C2,C3)-generated, and we also refer to (C1,C2,C3) as a generating triple for G.

Further, if H is a subgroup of G containing the fixed element z∈C3 above, we denote by ΣH⁢(C1,C2,C3) the total number of distinct ordered pairs (x,y)∈Ω such that 〈x,y〉≤H. The value of ΣH⁢(C1,C2,C3) is obtained as the sum of all structure constants ΔH⁢(c1,c2,c3), where the ci are conjugacy classes of H that fuse to Ci in G, i.e., ci⊆H∩Ci. The number of pairs (x,y)∈Ω generating a subgroup H of G will be denoted by ΣH*⁢(C1,C2,C3), and the centralizer of a representative of the conjugacy class C by CG⁢(C). A general conjugacy class of a proper subgroup H of G whose elements are of order n will be denoted by nx, reserving the notation nX for the case where H=G.

We remark that on occasion we shall abbreviate

ΔG⁢(C1,C2,C3),ΔG*⁢(C1,C2,C3),ΣH⁢(C1,C2,C3),ΣH*⁢(C1,C2,C3)

by Δ⁢(G), Δ*⁢(G), Σ⁢(H) and Σ*⁢(H), respectively, when the conjugacy classes C1,C2,C3 are understood from context.

The number of conjugates of a specified subgroup H of G containing a fixed element g is given by π⁢(g), where π=πH is the permutation character corresponding to the action of G on the conjugates of H, i.e., π is the induced character (1H)G (cf. [25]). As the stabilizer of H in this action is clearly NG⁢(H), in many cases, one can more easily compute the value π⁢(g) from the fusion map from NG⁢(H) into G in conjunction with Theorem 2.1 below. We emphasize that this is an especially useful strategy when the decomposition of π into irreducible characters is not known explicitly, or is not readily computable.

Theorem 2.1 (Ganief and Moori [21]).

Let G be a finite group, and let H be a subgroup of G containing a fixed element g such that gcd(o(g),|NG(H):H|)=1. Then the number of conjugates of H containing g is given by

π⁢(g)=∑i=1m|CG⁢(g)||CNG⁢(H)⁢(gi)|,

where π is the permutation character corresponding to the action of G on the conjugates of H, and g1,g2,…,gm are representatives of the NG⁢(H)-conjugacy classes that fuse to the G-class containing g.

Below we provide some very useful techniques for establishing non-generation.

Lemma 2.2 (Woldar [29]).

Let G be a finite group, and let x,y∈G. Suppose that Δ⁢(G)<|CG⁢(x⁢y)|, where Δ⁢(G)=ΔG⁢(l⁢X,m⁢Y,n⁢Z) with x∈l⁢X, y∈m⁢Y and x⁢y∈n⁢Z. Then CG⁢(〈x,y〉) is non-trivial.

Lemma 2.3 (Conder, Wilson and Woldar [10]).

Let G be a finite centerless group, and suppose lX, mY, nZ are G-conjugacy classes for which

Δ*⁢(G)=ΔG*⁢(l⁢X,m⁢Y,n⁢Z)<|CG⁢(n⁢Z)|.

Then Δ*⁢(G)=0, and therefore G is not (l⁢X,m⁢Y,n⁢Z)-generated.

We list all maximal subgroups of Fi24′ in Table 1. In Table 2, we indicate the fusion map from each maximal subgroup M into Fi24′, and we calculate the corresponding value of π⁢(z), where π is the permutation character (1M)Fi24′ and z∈M has prime order o⁢(z)≥7. Many of our computations relied heavily on the use of GAP [31], as well as certain subroutines provided by Ali, Al-Kadhi, Aljouiee and Ibrahim; see, e.g., [4]. As always, the ATLAS [11] served as an invaluable source of information, and we adopt its notation for conjugacy classes, maximal subgroups, etc.

Table 1

The maximal subgroups of Fi24′.

Group	Order
Fi23	218⁢.313⁢.52⁢.7.11.13.17.23
2.Fi22:2	219⁢.39⁢.52⁢.7.11.13
(3×O8+(3):3):2	213⁢.314⁢.52⁢.7.13
O10-⁢(2)	220⁢.36⁢.52⁢.7.11.17
37⋅O7⁢(3)	29⁢.316⁢.5.7.13
31+10:U5⁢(2):2	211⁢.316⁢.5.11
211⋅M24	221⁢.33⁢.5.7.11.23
22⋅U6⁢(2):S3	218⁢.37⁢.5.7.11
21+12⁢.3⁢U4⁢(3)⁢.22	221⁢.37⁢.5.7
33.[310].GL3⁢(3)	25⁢.316⁢.13
32+4+8.(A5×2⁢A4)⁢.2	26⁢.316⁢.5
(A4×O8+⁢(2)⁢.3)⁢.2	215⁢.37⁢.52⁢.7
He:2 (2 classes)	211⁢.33⁢.52⁢.73⁢.17
23+12.(L3⁢(2)×A6)	221⁢.33⁢.5.7
26+8.(S3×A8)	221⁢.33⁢.5.7
(32:2×G2(3)).2	28⁢.38⁢.7.13
(A5×A9):2	29⁢.35⁢.52⁢.7
A6×L2⁢(8):3	26⁢.33⁢.5.72
7:6×A7	24⁢.33⁢.5.72
U3⁢(3):2 (2 classes)	26⁢.33⁢.7
L2⁢(13):2 (2 classes)	23⁢.3.7.13
29:14	2.7.29

Table 2

Partial fusion maps into Fi24′.

Fi23-class

3⁢a

3⁢b

3⁢c

3⁢d

5⁢a

7⁢a

11⁢a

13⁢a

13⁢b

17⁢a

23⁢a

23⁢b

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢D

5⁢A

7⁢A

11⁢A

13⁢A

17⁢A

23⁢A

23⁢B

π⁢(z)

2.Fi22:2-class

3⁢a

3⁢b

3⁢c

3⁢d

5⁢a

7⁢a

11⁢a

13⁢a

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢D

5⁢A

7⁢A

11⁢A

13⁢A

π⁢(z)

105

(3×O8+(3):3):2-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

3⁢h

3⁢i

3⁢j

3⁢k

3⁢l

→Fi24′

3⁢B

3⁢C

3⁢A

3⁢B

3⁢D

3⁢C

3⁢D

3⁢A

3⁢D

3⁢A

3⁢B

(3×O8+(3):3):2-class

3⁢m

3⁢n

3⁢o

3⁢p

3⁢q

3⁢r

3⁢s

3⁢t

3⁢u

7⁢a

13⁢a

13⁢b

→Fi24′

3⁢C

3⁢D

3⁢C

3⁢D

3⁢E

7⁢A

13⁢A

π⁢(z)

O10-⁢(2)-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

5⁢a

5⁢b

7⁢a

11⁢a

11⁢b

17⁢a

17⁢b

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢D

5⁢A

7⁢A

11⁢A

17⁢A

π⁢(z)

37⋅O7⁢(3)-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

3⁢h

3⁢i

3⁢j

3⁢k

3⁢l

→Fi24′

3⁢B

3⁢C

3⁢A

3⁢D

3⁢C

3⁢D

3⁢A

3⁢B

3⁢D

3⁢C

3⁢B

37⋅O7⁢(3)-class

3⁢m

3⁢n

3⁢o

3⁢p

3⁢q

3⁢r

3⁢s

3⁢t

5⁢a

7⁢a

13⁢a

13⁢b

→Fi24′

3⁢C

3⁢D

3⁢A

3⁢E

3⁢D

5⁢A

7⁢B

13⁢A

π⁢(z)

31+10:U5⁢(2):2-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

3⁢h

3⁢i

3⁢j

3⁢k

3⁢l

→Fi24′

3⁢B

3⁢A

3⁢B

3⁢C

3⁢D

3⁢A

3⁢B

3⁢C

3⁢D

3⁢B

3⁢C

3⁢D

31+10:U5⁢(2):2-class

3⁢m

3⁢n

3⁢o

3⁢p

3⁢q

3⁢r

3⁢s

3⁢t

3⁢u

3⁢v

5⁢a

11⁢a

→Fi24′

3⁢D

3⁢A

3⁢D

3⁢E

3⁢C

3⁢D

3⁢E

5⁢A

11⁢A

π⁢(z)

211⋅M24-class

3⁢a

3⁢b

5⁢a

7⁢a

7⁢b

11⁢a

23⁢a

23⁢b

→Fi24′

3⁢C

3⁢E

5⁢A

7⁢A

11⁢A

23⁢A

23⁢B

π⁢(z)

210

22⋅U6⁢(2):S3-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

5⁢a

7⁢a

11⁢a

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢B

3⁢C

3⁢D

3⁢E

5⁢A

7⁢A

11⁢A

π⁢(z)

105

21+12⁢.3⁢U4⁢(3)⁢.22

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

5⁢a

7⁢a

→Fi24′

3⁢C

3⁢D

3⁢B

3⁢A

3⁢D

3⁢C

3⁢E

5⁢A

7⁢B

π⁢(z)

(A4×O8+⁢(2)⁢.3)⁢.2-class

3⁢a

3⁢b

3⁢c/g

3⁢d

3⁢e

3⁢f

3⁢h

3⁢i

3⁢j/k

3⁢l/m

5⁢a

7⁢a

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢A

3⁢D

3⁢A

3⁢D

3⁢E

5⁢A

7⁢A

π⁢(z)

He:2-class

3⁢a

3⁢b

5⁢a

7⁢a

7⁢b

7⁢c

17⁢a

→Fi24′

3⁢C

3⁢E

5⁢A

7⁢A

7⁢B

17⁢A

π⁢(z)

23+12.(L3⁢(2)×A6)-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

5⁢a

5⁢b

7⁢a

7⁢b

→Fi24′

3⁢C

3⁢A

3⁢E

3⁢D

3⁢E

5⁢A

7⁢A

π⁢(z)

26+8.(S3×A8)-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

5⁢a

7⁢a

7⁢b

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢D

5⁢A

7⁢A

π⁢(z)

210

(32:2×G2(3)).2-class

3⁢a

3⁢b

3⁢c/g

3⁢d

3⁢e/f

3⁢g/h

3⁢i/j

3⁢k/l

3⁢m/n

3⁢o/p

7⁢a

13⁢a

→Fi24′

3⁢B

3⁢D

3⁢C

3⁢A

3⁢D

3⁢E

7⁢A

13⁢A

π⁢(z)

(A5×A9):2-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

5⁢a

5⁢b

5⁢c

7⁢a

→Fi24′

3⁢A

3⁢B

3⁢C

3⁢A

3⁢C

3⁢D

5⁢A

7⁢A

π⁢(z)

7:6×A7-class

3⁢a

3⁢b

3⁢c

3⁢d

3⁢e

3⁢f

3⁢g

3⁢h

5⁢a

7⁢a

7⁢b

7⁢c

7⁢d

7⁢e

→Fi24′

3⁢A

3⁢E

3⁢C

3⁢D

3⁢E

3⁢C

3⁢D

3⁢E

5⁢A

7⁢B

7⁢A

π⁢(z)

721

U3⁢(3):2-class

2⁢a

2⁢b

3⁢a⁢c

3⁢b

7⁢a

→Fi24′

2⁢B

3⁢D

3⁢E

7⁢B

π⁢(z)

294

L2⁢(13):2-class

3⁢a

7⁢a

7⁢b

7⁢c

13⁢a

→Fi24′

3⁢D

7⁢B

13⁢A

vπ⁢(z)

441

3 (3,q,r)-Generation of Fi24′

The Fischer’s group Fi24′ is a sporadic simple group of order

1255205709190661721292800=221⋅316⋅52⋅73⋅11⋅13⋅17⋅23⋅29.

The group Fi24′ was discovered by Bernd Fischer [18] in 1971, resulting from his classification of finite 3-transposition groups with no non-trivial normal solvable subgroups. It is the largest of his sporadic simple groups, being the commutator subgroup of the 3-transposition group Fi24=Aut⁡(Fi24′). It has 108 conjugacy classes in total, including two classes of involutions (viz. 2⁢A,2⁢B) and five classes of elements of order 3 (viz. 3A,3B,3C,3D,3E). Linton & Wilson [26] investigated the subgroup structure of Fi24′ and classified all maximal subgroups of Fi24′ as well as those of its automorphism group Fi24.

We now proceed with a case-by-case analysis of all (3,q,r)-generations of Fi24′, where r∈{7,11,13,17,23,29}.

3.1 The case r=7

Since there are five classes of elements of order 3, a single class of elements of order 5, and two classes of elements of order 7, we must consider 10 distinct triples of conjugacy classes in Fi24′.

We begin with the Fi24′-classes 7⁢A, which requires a much more delicate treatment than that devoted to the class 7⁢B.

Lemma 3.4.

Let G=Fi24 (so that G′=Fi24′), and fix z∈7⁢A. Then CG⁢(z)=〈z〉×H with H≅S7. Identifying H with S7, we have that the 21 transpositions (a,b)∈H are of G-class 2⁢C, the 105 involutions of type (a,b)⁢(c,d)∈H are of G-class 2⁢A, and the 105 involutions of type (a,b)⁢(c,d)⁢(e,f)∈H are of G-class 2⁢D. Finally, the 70 elements of type (a,b,c)∈H are of G-class 3⁢A.

Proof.

As CG′⁢(z)≅7×A7, we see at once that elements of type (a,b)∈H and those of type (a,b)⁢(c,d)⁢(e,f)∈H must lie outside G′. Thus elements of each type are either of G-class 2⁢C or 2⁢D. Since (1,2)∈H centralizes z⁢(3,4,5,6,7)∈CG⁢(z), we see that w=z⁢(1,2)⁢(3,4,5,6,7) is of order 70 and w35=(1,2). But this can only occur if (1,2)∈2⁢C.

Note that despite the fact that (a,b)∈H and (a,b)⁢(c,d)⁢(e,f)∈H are obviously not conjugate in H, they could be conjugate in G. However, by character computation, we know that z lies in exactly 21 G′-conjugates of F≅Fi23 (see Table 2). Thus z lies in exactly 21 G-conjugates of CG⁢(t)=〈t〉×F, where t∈2⁢C. These 21 conjugates correspond to the 21 elements (a,b)∈2⁢C; hence there can be no other elements in CG⁢(z) of type 2⁢C. Thus elements of type (a,b)⁢(c,d)⁢(e,f)∈H must be in the G-class 2⁢D.

As elements of the form (a,b)⁢(c,d) are in H′≅A7, they must be in G′, and thus they are either of class 2⁢A or 2⁢B. However, nothing in class 2⁢B commutes with z∈7⁢A; hence (a,b)⁢(c,d)∈2⁢A. Finally, elements of type (a,b,c) are of G-class 3⁢A because v=z⁢(1,2,3)⁢(4,5,6,7) is of order 84 with v28=(1,2,3). But this can only occur if (1,2,3)∈3⁢A. (Alternatively, one could argue that the product of two non-commuting involutions of Fi24-type 2⁢C must be of Fi24-type 3⁢A, a known property of the 3-transposition class 2⁢C.) ∎

Lemma 3.5.

With notation as in Lemma 3.4, let S be any set of transpositions in H. Then the subgroup 〈S〉<H generated by S is isomorphic to one of the following groups: S7, S6, S5, S5×S2, S4, S4×S2, S4×S3, S3, S3×S2, S3×S22, S32, S2, S22, S23. Moreover, the numbers of such subgroups of H that so arise are indicated as follows:

S7	S6	S5	S5×S2	S4	S4×S2	S4×S3
1	7	21	21	35	105	35

S3	S3×S2	S3×S22	S32	S2	S22	S23
35	210	105	70	21	105	105

Proof.

This is an easy counting exercise, which we leave to the reader. ∎

Our next goal is to determine the centralizer CG⁢(S) of each subgroup 〈S〉≤H≅S7, as well as its contribution Σ⁢(CG⁢(S)) to the structure constant Δ⁢(Fi24)=269157 of type (3⁢A,5⁢A,7⁢A). The results are shown in Table 3. Note that if S and T are sets of transpositions in H≅S7 with 〈S〉≅〈T〉, then 〈S〉 and 〈T〉 are conjugate in H; hence they are conjugate in G=Fi24.

Table 3

Contributions of CG⁢(S) toward ΔFi24⁢(3⁢A,5⁢A,7⁢A).

〈S〉	CG⁢(S)	Σ⁢(CG⁢(S))
S7	7:6	0
S6	L2⁢(8):3	0
S5	S9	7
S5×S2	2×S7	7
S4	O8+⁢(2):S3	63
S4×S2	2×S6⁢(2)	7
S4×S3	U3⁢(3):2	0
S3	O8+⁢(3):S3	1680
S3×S2	2×O7⁢(3)	336
S3×S22	2×2⁢U4⁢(3)	336
S32	G2⁢(3)	0
S2	2×Fi23	33117
S22	2×2⁢F⁢i22	4977
S23	2×22⁢U6⁢(2)	1281

In Table 4, we indicate the number of ways a fixed subgroup 〈S〉 of H≅S7 of given isomorphism type can be generated by exactly n transpositions. This information is important to us since n is also the precise number of transposition-centralizers whose intersection gives the corresponding group CG⁢(S). (Indeed, CG⁢(S)=⋂s∈SCG⁢(s).) We shall use this information later on, when we invoke the principle of inclusion-exclusion to prove that Fi24′ cannot be (3⁢A,5⁢A,7⁢A)-generated.

Table 4

Number of ways a specified subgroup of H can be generated by a set S of n transpositions (1≤n≤11).

〈S〉	1	2	3	4	5	6	7	8	9	10	11
S7	0	0	0	0	0	17227	68295	156555	258125	331506	343140
S6	0	0	0	0	1296	3660	5700	6165	4945	2997	1365
S5	0	0	0	125	222	205	120	45	10	1	0
S5×S2	0	0	0	0	125	222	205	120	45	10	1
S4	0	0	16	15	6	1	0	0	0	0	0
S4×S2	0	0	0	16	15	6	1	0	0	0	0
S4×S3	0	0	0	0	48	61	33	9	1	0	0
S3	0	3	1	0	0	0	0	0	0	0	0
S3×S2	0	0	3	1	0	0	0	0	0	0	0
S3×S22	0	0	0	3	1	0	0	0	0	0	0
S32	0	0	0	9	6	1	0	0	0	0	0
S2	1	0	0	0	0	0	0	0	0	0	0
S22	0	1	0	0	0	0	0	0	0	0	0
S23	0	0	1	0	0	0	0	0	0	0	0

The reader will observe that the table stops at n=11. This is because larger values of n fail to contribute to Δ⁢(Fi24). We also remark that if one multiplies each entry in Table 4 by the number of subgroups 〈S〉≤H of the indicated isomorphism type, then the resulting column sums will be (21n), 1≤n≤11. This provides a valuable check on the accuracy of Table 4.

Proposition 3.6.

The group Fi24′ is not (3⁢A,5⁢A,7⁢A)-generated.

Proof.

We proceed by inclusion-exclusion, but first we must lay some groundwork. Note that all computations will proceed in Fi24 rather than in Fi24′.

We start by determining the precise number of CG⁢(z)-conjugates of the centralizer CG⁢(t)≅2×Fi23 where t∈H≤CG⁢(z), as well as the contribution to Δ⁢(Fi24) coming from all (3⁢A,5⁢A,7⁢A)-subgroups contained in these conjugates. In subsequent steps, we do this for intersections of two CG⁢(z)-conjugates of CG⁢(t) and then for intersections of three such conjugates and so on, ultimately reaching eleven.

All data used in the following computations may be gleaned from Tables 3 and 4 as well as Lemma 3.5. We will henceforth regard the four rows of Table 4 headed by S7, S6, S4×S3 and S32 as irrelevant since their corresponding centralizers 7:6, L2⁢(8):3, U3⁢(3):2 and G2⁢(3) contribute zero to Δ⁢(Fi24).

One conjugate. There are 21 CG⁢(z)-conjugates of CG⁢(t)≅2×Fi23 and each contributes 33117 to Δ⁢(Fi24). This gives a total contribution of 21×33117=695457.

Two conjugates. There are 210 intersections of two CG⁢(z)-conjugates of CG⁢(t) and these are of two types: (1) 3×35=105 intersections isomorphic to O8+⁢(3):S3, giving a contribution of 105×1680=176400, and (2) 1×105=105 intersections isomorphic to 2×2⁢F⁢i22, giving a contribution of 105×4977=522585. Thus the total contribution to Δ⁢(Fi24) coming from intersections of two conjugates is 176400+522585=698985.

Three conjugates. Intersections here are of four types: (1) 16×35=560 intersections isomorphic to O8+⁢(2):S3, giving a contribution of 560×63=35280, (2) 1×35=35 intersections isomorphic to O8+⁢(3):S3, giving a contribution of 35×1680=58800, (3) 3×210=630 intersections isomorphic to 2×O7⁢(3), giving a contribution of 630×336=211680, and (4) 1×105=105 intersections isomorphic to 2×22⁢U6⁢(2), giving a contribution of 105×1281=134505. Thus the total contribution to Δ⁢(Fi24) coming from intersections of three conjugates is 35280+58800+211680+134505=440265.

Four conjugates. Relevant intersections are of five types: (1) 125×21=2625 intersections isomorphic to S9, giving a contribution of 2625×7=18375, (2) 15×35=525 intersections isomorphic to O8+⁢(2):S3, giving a contribution of 525×63=33075, (3) 16×105=1680 intersections isomorphic to 2×S6⁢(2), giving a contribution of 1680×7=11760, (4) 1×210=210 intersections isomorphic to 2×O7⁢(3), giving a contribution of 210×336=70560, (5) 3×105=315 intersections isomorphic to 2×2⁢U4⁢(3), giving a contribution of 315×336=105840. Thus the total contribution to Δ⁢(Fi24) coming from intersections of four conjugates is 18375+33075+11760+70560+105840=239610.

Five conjugates. Relevant intersections are of five types: (1) 222×21=4662 intersections isomorphic to S9, giving a contribution of 4662×7=32634, (2) 125×21=2625 intersections isomorphic to 2×S7, giving a contribution of 2625×7=18375, (3) 6×35=210 intersections isomorphic to O8+⁢(2):S3, giving a contribution of 210×63=13230, (4) 15×105=1575 intersections isomorphic to 2×S6⁢(2), giving a contribution of 1575×7=11025, and (5) 1×105=105 intersections isomorphic to 2×2⁢U4⁢(3), giving a contribution of 105×336=35280. Thus the total contribution to Δ⁢(Fi24) coming from intersections of five conjugates is 32634+18375+13230+11025+35280=110544.

Six conjugates. Relevant intersections are of four types: (1) 205×21=4305 intersections isomorphic to S9, giving a contribution of 4305×7=30135, (2) 222×21=4662 intersections isomorphic to 2×S7, giving a contribution of 4662×7=32634, (3) 1×35=35 intersections isomorphic to O8+⁢(2):S3, giving a contribution of 35×63=2205, and (4) 6×105=630 intersections isomorphic to 2×S6⁢(2), giving a contribution of 630×7=4410. Thus the total contribution to Δ⁢(Fi24) coming from intersections of six conjugates is 30135+32634+2205+4410=69384.

Seven conjugates. Here relevant intersections are of three types: (1) 120×21=2520 intersections isomorphic to S9, giving a contribution of 2520×7=17640, (2) 205×21=4305 intersections isomorphic to 2×S7, giving a contribution of 4305×7=30135, and (3) 1×105=105 intersections isomorphic to 2×S6⁢(2), giving a contribution of 105×7=735. Thus the total contribution to Δ⁢(Fi24) coming from intersections of seven conjugates is 17640+30135+735=48510.

Eight conjugates. There are just two types of relevant intersections in this case: (1) 45×21=945 intersections isomorphic to S9, giving a contribution of 945×7=6615, and (2) 120×21=2520 intersections isomorphic to 2×S7, giving a contribution of 2520×7=17640. Thus the total contribution to Δ⁢(Fi24) coming from intersections of eight conjugates is 6615+17640=24255.

Nine conjugates. In this case, there are again two types of relevant intersections: (1) 10×21=210 intersections isomorphic to S9, giving a contribution of 210×7=1470, and (2) 45×21=945 intersections isomorphic to 2×S7, giving a contribution of 945×7=6615. Thus the total contribution to Δ⁢(Fi24) coming from intersections of nine conjugates is 1470+6615=8085.

Ten conjugates. Once more, there are only two types of relevant intersections: (1) 1×21=21 intersections isomorphic to S9, giving a contribution of 21×7=147, and (2) 10×21=210 intersections isomorphic to 2×S7, giving a contribution of 210×7=1470. Thus the total contribution to Δ⁢(Fi24) coming from intersections of ten conjugates is 147+1470=1617.

Eleven conjugates. There is just one type of relevant intersection in this case, namely 21 intersections isomorphic to 2×S7, giving a contribution of 21×7=147.

We are now prepared to invoke the principle of inclusion-exclusion to account for all (3⁢A,5⁢A,7⁢A)-triples of Fi24 that lie in at least one conjugate of CG⁢(t), t∈CG⁢(z). This involves adding all contributions to Δ⁢(Fi24) coming from intersections of an odd number of conjugates, from which we subtract all contributions coming from intersections of an even number of conjugates. Specifically, this gives

695457-698985+440265-239610+110544-69384+48510-24255+8085-1617+147=269157,

which is the precise value of Δ⁢(Fi24). We conclude that every (3⁢A,5⁢A,7⁢A)-subgroup of Fi24 must lie in some CG⁢(z)-conjugate of CG⁢(t)≅Fi23×2, whence Fi24′ cannot be (3⁢A,5⁢A,7⁢A)-generated. ∎

Proposition 3.7.

Fi24′ is (3⁢B,5⁢A,7⁢A)-generated.

Proof.

Our methodology is identical to that of Proposition 3.6, so we may avoid repeating many of the details. The relevant contributions Σ⁢(CG⁢(S)) to the structure constant ΔFi24⁢(3⁢B,5⁢A,7⁢A)=3004533 are provided in Table 5. Note that we may regard the five rows of Table 4 headed by S7, S6, S5×S2, S4×S3 and S32 as irrelevant since their corresponding centralizers 7:6, L2⁢(8):3, 2×S7, U3⁢(3):2 and G2⁢(3) contribute zero to Δ⁢(Fi24).

Table 5

Contributions of CG⁢(S) toward ΔFi24⁢(3⁢B,5⁢A,7⁢A).

〈S〉	CG⁢(S)	Σ⁢(CG⁢(S))
S7	7:6	0
S6	L2⁢(8):3	0
S5	S9	28
S5×S2	2×S7	0
S4	O8+⁢(2):S3	903
S4×S2	2×S6⁢(2)	77
S4×S3	U3⁢(3):2	0
S3	O8+⁢(3):S3	18648
S3×S2	2×O7⁢(3)	1988
S3×S22	2×2⁢U4⁢(3)	56
S32	G2⁢(3)	0
S2	2×Fi23	285215
S22	2×2⁢F⁢i22	28469
S23	2×22⁢U6⁢(2)	1071

One conjugate. The 21 conjugates of 2×Fi23 each contribute 285215, yielding a total contribution of 21×285215=5989515.

Two conjugates. The 105 intersections of type O8+⁢(3):S3 (meaning, each intersection is isomorphic to O8+⁢(3):S3) account for 105×18648=1958040, while the 105 intersections of type 2×2⁢F⁢i22 account for a total contribution of 105×28469=2989245. Thus the total contribution is 4947285.

Three conjugates. Here we get respective contributions of 560×903=505680 (of type O8+⁢(2):S3), 35×18648=652680 (of type O8+⁢(3):S3), 630×1988=1252440 (of type 2×O7⁢(3)) and 105×1071=112455 (of type 2×22⁢U6⁢(2)). The total contribution is therefore 2523255.

Four conjugates. Here we get contributions of 2625×28=73500 (of type S9), 525×903=474075 (of type O8+⁢(2):S3), 1680×77=129360 (of type 2×S6⁢(2)), 210×1988=417480 (of type 2×O7⁢(3)) and 315×56=17640 (of type 2×2⁢U4⁢(3)). Thus the total contribution is 1112055.

Five conjugates. We get contributions of 4662×28=130536 (of type S9), 210×903=189630 (of type O8+⁢(2):S3), 1575×77=121275 (of type 2×S6⁢(2)) and 105×56=5880 (of type 2×2⁢U4⁢(3)). Thus the total contribution here is 447321.

Six conjugates. Here we get contributions of 4305×28=120540 (of type S9), 35×903=31605 (of type O8+⁢(2):S3) and 630×77=48510 (of type 2×S6⁢(2)). This yields a total contribution of 200655.

Seven conjugates. Contributions here are 2520×28=70560 (of type S9) and 105×77=8085 (of type 2×S6⁢(2)), thus yielding a total contribution of 78645.

Eight conjugates. The total contribution is 945×28=26460 (of type S9).

Nine conjugates. The total contribution is 210×28=5880 (of type S9).

Ten conjugates. The total contribution is 21×28=588 (of type S9).

Eleven conjugates. Here there is no contribution since the only possibility for the intersection of 11 conjugates is 2×S7.

Using the principle of inclusion-exclusion, we may now compute the total contribution of the groups CG⁢(S) to Δ⁢(Fi24):

5989515-4947285+2523255-1112055+447321-200655+78645-26460+5880-588=2757573.

As this value is strictly less than the structure constant Δ⁢(Fi24)=3004533, the proof is complete. ∎

Proposition 3.8.

Fi24′ is (3⁢X,5⁢A,7⁢A)-generated for all X∈{C,D,E}.

Proof.

Any maximal subgroup of Fi24′ with order divisible by 3⋅5⋅7 must be isomorphic to one of

Fi23,2.Fi22:2,(3×O8+(3):3):2,O10-(2),37⋅O7(3),211⋅M24,22⋅U6(2):S3,21+12.U4(3).22,(A4×O8+(2).3).2,He:2(2 classes),23+12.(L3(2)×A6),26+8.(S3×A8),(A5×A9):2,A6×L2⁢(8):3 or 7:6×A7.

Consider first the triple (3⁢C,5⁢A,7⁢A) with structure constant

ΔFi24′⁢(3⁢C,5⁢A,7⁢A)=534146403.

Among the maximal subgroups of Fi24′ with order divisible by 105, both 37⋅O7⁢(3) and 21+12⁢.3⁢U4⁢(3)⁢.22 fail to meet the Fi24′-conjugacy class 7⁢A. Moreover, none of 23+12.(L3⁢(2)×A6), A6×L2⁢(8):3, 7:6×A7 contributes to ΔFi24′⁢(3⁢C,5⁢A,7⁢A) since

Σ23+12.(L3⁢(2)×A6)⁢(3⁢C,5⁢A,7⁢A)=ΣA6×L2⁢(8):3⁢(3⁢C,5⁢A,7⁢A)=Σ7:6×A7⁢(3⁢C,5⁢A,7⁢A)=0.

For each of the remaining maximal subgroups M of Fi24′, we obtain their respective contributions to the structure constant ΔFi24′⁢(3⁢C,5⁢A,7⁢A) from their corresponding character tables and the fusion maps given in Table 2. Specifically, we have

ΣFi23⁢(3⁢C,5⁢A,7⁢A)=15934191,

Σ2.Fi22⁣:2⁢(3⁢C,5⁢A,7⁢A)=742455,

Σ(3×O8+(3):3):2⁢(3⁢C,5⁢A,7⁢A)=219408,

ΣO10-⁢(2)⁢(3⁢C,5⁢A,7⁢A)=302652,

Σ211⋅M24⁢(3⁢C,5⁢A,7⁢A)=83328,

Σ22⋅U6⁢(2):S3⁢(3⁢C,5⁢A,7⁢A)=59787,

Σ(A4×O8+⁢(2)⁢.3)⁢.2⁢(3⁢C,5⁢A,7⁢A)=2709,

ΣHe:2⁢(3⁢C,5⁢A,7⁢A)=1470,

Σ26+8.(S3×A8)⁢(3⁢C,5⁢A,7⁢A)=6720,

Σ(A5×A9):2⁢(3⁢C,5⁢A,7⁢A)=98.

Table 2 also provides us with the number πM⁢(z) of conjugates of M containing a fixed element z of Fi24′-type 7⁢A. For convenience, we replicate that portion of Table 2 relevant to the Fi24′-class 7⁢A as follows:

M	Fi23	2.Fi22:2	(3×O8+(3):3):2	O10-⁢(2)	211⋅M24	22⋅U6⁢(2):S3
πM⁢(z)	21	105	35	42	210	105

M	(A4×O8+⁢(2)⁢.3)⁢.2	He:2 (2 classes)	26+8.(S3×A8)	(A5×A9):2
πM⁢(z)	35	15	210	21

From this information, we obtain

ΔFi24′*⁢(3⁢C,5⁢A,7⁢A)≥ΔFi24′(3C,5A,7A)-21Σ(Fi23)-105Σ(2.Fi22:2)-35Σ((3×O8+(3):3):2)-42Σ(O10-(2))-210Σ(211⋅M24)-105Σ(22⋅U6(2):S3)-35Σ((A4×O8+(2).3).2)-30Σ(He:2)-210Σ(26+8.(S3×A8))-21Σ((A5×A9):2)=534146403-21⁢(15934191)-105⁢(742455)-35⁢(219408)-42⁢(302652)-210⁢(83328)-105⁢(59787)-35⁢(2709)-30⁢(1470)-210⁢(6720)-21⁢(98)>0,

proving that (3⁢C,5⁢A,7⁢A) is a generating triple of Fi24′.

The argument is similar for the triples (3⁢D,5⁢A,7⁢A) and (3⁢E,5⁢A,7⁢A). We calculate the relevant structure constants to be ΔFi24′⁢(3⁢D,5⁢A,7⁢A)=11518064460 and ΔFi24′⁢(3⁢E,5⁢A,7⁢A)=40223327760. The inequalities below reflect only those maximal subgroups of Fi24′ that contribute to these structure constants, as well as the number of conjugates in which a fixed element of type 7⁢A is contained. As before, we rely heavily on fusion maps (see Table 2) for each relevant subgroup of Fi24′.

ΔFi24′*⁢(3⁢D,5⁢A,7⁢A)≥ΔFi24′(3D,5A,7A)-21Σ(Fi23)-105Σ(2.Fi22:2)-35Σ((3×O8+(3):3):2)-42Σ(O10-(2))=11518064460-21⁢(167942880)-105⁢(2592324)-35⁢(660240)-42⁢(1155000)>0

ΔFi24′*⁢(3⁢E,5⁢A,7⁢A)≥ΔFi24′(3E,5A,7A)-210Σ(211⋅M24)-30Σ(He:2)=40223327760-210⁢(190848)-30⁢(31752)>0.

Thus generation of Fi24′ by each of the triples (3⁢D,5⁢A,7⁢A) and (3⁢E,5⁢A,7⁢A) is also established. ∎

Proposition 3.9.

Fi24′ is not (3⁢A,5⁢A,7⁢B)-generated.

Proof.

The result follows immediately from Lemma 2.3 since

ΔFi24′⁢(3⁢A,5⁢A,7⁢B)=1470

and |CFi24′⁢(z)|=2058 for z∈7⁢B. ∎

Proposition 3.10.

Fi24′ is (3⁢X,5⁢A,7⁢B)-generated for all X∈{B,C,D,E}.

Proof.

The proof may divided into two cases:

Case 1. X∈{B,D}: Here

ΔFi24′⁢(3⁢B,5⁢A,7⁢B)=963830,

ΔFi24′⁢(3⁢D,5⁢A,7⁢B)=8832037340.

The only maximal subgroups of Fi24′ that meets all required conjugacy classes are (in both cases) 37⋅O7⁢(3) and 21+12⁢.3⁢U4⁢(3)⁢.22. From Table 2, it follows that

ΔFi24′*⁢(3⁢B,5⁢A,7⁢B)≥ΔFi24′⁢(3⁢B,5⁢A,7⁢B)-49⁢Σ⁢(37⋅O7⁢(3))-49Σ(21+12.U4(3).22)=963830-49⁢(854)-49⁢(182)>0,

ΔFi24′*⁢(3⁢D,5⁢A,7⁢B)≥ΔFi24′⁢(3⁢D,5⁢A,7⁢B)-49⁢Σ⁢(37⋅O7⁢(3))-49Σ(21+12.U4(3).22)=8832037340-49⁢(226436)-49⁢(18760)>0.

Thus both (3⁢B,5⁢A,7⁢B) and (3⁢D,5⁢A,7⁢B) are generating triples for Fi24′.

Case 2. X∈{C,E}: Here

ΔFi24′⁢(3⁢C,5⁢A,7⁢B)=62154344,

ΔFi24′⁢(3⁢E,5⁢A,7⁢B)=34226708544.

In each case, the relevant maximal subgroups, their respective number of conjugates, and the contributions coming from these conjugates are all indicated in the computations below (cf. Table 2; further note that Σ7:6×A7⁢(3⁢C,5⁢A,7⁢B)=0):

ΔFi24′*⁢(3⁢C,5⁢A,7⁢B)≥ΔFi24′⁢(3⁢C,5⁢A,7⁢B)-49⁢Σ37⋅O7⁢(3)⁢(3⁢C,5⁢A,7⁢B)-49⁢Σ21+12⁢.3⁢U4⁢(3)⁢.22⁢(3⁢C,5⁢A,7⁢B)-44⁢ΣHe:2⁢(3⁢C,5⁢A,7⁢B)=62154344-49⁢(4004)-49⁢(5376)-44⁢(343)>0,

ΔFi24′*⁢(3⁢E,5⁢A,7⁢B)≥ΔFi24′⁢(3⁢E,5⁢A,7⁢B)-49⁢Σ37⋅O7⁢(3)⁢(3⁢E,5⁢A,7⁢B)-49⁢Σ21+12⁢.3⁢U4⁢(3)⁢.22⁢(3⁢E,5⁢A,7⁢B)-44⁢ΣHe:2⁢(3⁢E,5⁢A,7⁢B)-14⁢Σ7:6×A7⁢(3⁢E,5⁢A,7⁢B)=34226708544-49⁢(249480)-49⁢(64512)-44⁢(45962)-14⁢(56)>0.

Thus both (3⁢C,5⁢A,7⁢B) and (3⁢E,5⁢A,7⁢B) are generating triples for Fi24′. ∎

3.2 The case r=11

Here we need only consider maximal subgroups of Fi24′ with order divisible by 11, i.e., Fi23, 2.Fi22:2, 211.M24, O10-⁢(2), 31+10:U5⁢(2):2, 22.U6(2):S3.

3.2.1 The subcase q=5

Proposition 3.11.

Fi24′ is (3⁢X,5⁢A,11⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

It is evident from Table 2 that among all maximal subgroups of Fi24′ of order divisible by 11, the group 211⋅M24 fails to meet the Fi24′-classes 3⁢A, 3⁢B, 3⁢D. Similarly, each of Fi23, 2.Fi22:2, 31+10:U5⁢(2):2 fails to meet the Fi24′-class 3⁢E.

Of the remaining maximal subgroups, we see from Table 2 that a fixed element of Fi24′-type 11⁢A is contained in 3, 3, 8, 4, 4, 1 conjugates of Fi23, 2.Fi22:2, O10-⁢(2), 31+10:U5⁢(2):2, 211⋅M24, 22⋅U6⁢(2):S3, respectively. Thus we compute

ΔFi24′*⁢(3⁢A,5⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-4Σ(31+10:U5(2):2)-Σ(22⋅U6(2):S3)=116413-3⁢(22407)-3⁢(5225)-8⁢(33)-4⁢(2563)-1419>0,

ΔFi24′*⁢(3⁢B,5⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-4Σ(31+10:U5(2):2)-Σ(22⋅U6(2):S3)=494241-3⁢(94611)-3⁢(16137)-8⁢(946)-4⁢(4209)-1683>0,

ΔFi24′*⁢(3⁢C,5⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-4Σ(31+10:U5(2):2)-6Σ(211⋅M24)-Σ(22⋅U6(2):S3)=193111215-3⁢(7306101)-3⁢(506055)-8⁢(227975)-4⁢(104115)-6⁢(50160)-40821>0,

ΔFi24′*⁢(3⁢D,5⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-4Σ(31+10:U5(2):2)-Σ(22⋅U6(2):S3)=9594861914-3⁢(169063092)-3⁢(2729166)-8⁢(1219416)-4⁢(391556)-0>0,

ΔFi24′*⁢(3⁢E,5⁢A,11⁢A)≥Δ(Fi24′)-4Σ(31+10:U5(2):2)-6Σ(211⋅M24)-Σ(22⋅U6(2):S3)=39433451640-4⁢(1138698)-6⁢(234784)-0>0.

Hence all claimed generations are confirmed. ∎

3.2.2 The subcase q=7

Proposition 3.12.

Fi24′ is (3⁢X,7⁢Y,11⁢A)-generated for all X∈{A,B,C,D,E} and Y∈{A,B}.

Proof.

First we consider the case Y=A. From Table 1, we see that the only maximal subgroups that could potentially contribute to the relevant structure constants of Fi24′ are Fi23, 2.Fi22:2, O10-⁢(2), 31+10:U5⁢(2):2, 211⋅M24, 22⋅U6⁢(2):S3. However, from the fusion maps in Table 2, we observe that the Fi24′-classes 3⁢A, 3⁢B, 3⁢D fail to meet 211⋅M24, while 3⁢E fails to meet each of Fi23, 2.Fi22:2, O10-⁢(2). We compute

ΔFi24′*⁢(3⁢A,7⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-Σ(22⋅U6(2):S3)=2790051-3⁢(425161)-3⁢(49247)-8⁢(0)-4389>0,

ΔFi24′*⁢(3⁢B,7⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-Σ(22⋅U6(2):S3)=35104443-3⁢(3501355)-3⁢(285615)-8⁢(2574)-7095>0,

ΔFi24′*⁢(3⁢C,7⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-6Σ(211⋅M24)-Σ(22⋅U6(2):S3)=6438342537-3⁢(166717683)-3⁢(6014745)-8⁢(369468)-6⁢(247808)-168795>0,

ΔFi24′*⁢(3⁢D,7⁢A,11⁢A)≥Δ(Fi24′)-3Σ(Fi23)-3Σ(2.Fi22:2)-8Σ(O10-(2))-Σ(22⋅U6(2):S3)=477391185414-3⁢(5095787004)-3⁢(42283890)-8⁢(2342835)-0>0,

ΔFi24′*⁢(3⁢E,7⁢A,11⁢A)≥Δ(Fi24′)-6Σ(211⋅M24)-Σ(22⋅U6(2):S3)=1893954555240-6⁢(1250304)-0>0.

This completes the proof of the proposition for the case Y=A.

The case Y=B is much simpler. This is because the Fi24′-class 7⁢B fails to meet every maximal subgroup of Fi24′ with order divisible by 11. As

ΔFi24′*⁢(3⁢X,7⁢B,11⁢A)>0

for all X∈{A,B,C,D,E}, all asserted generations occur. ∎

3.3 The case r=13

We need only consider maximal subgroups of Fi24′ with order divisible by 13, i.e.,

Fi23,2.Fi22:2,(3×O8+(3):3):2,37⋅O7(3),33.[310].GL3(3),(32:2×G2(3)).2,L2(13):2(2 classes).

However, we may eliminate 33.[310].GL3⁢(3) from consideration since its order is divisible only by the primes 2, 3 and 13. We provide Table 6 below as an additional aid to the reader.

Table 6

Structure constants and contributions from maximal subgroups (r=13).

	qY	(3⁢A,q⁢Y,13⁢A)	(3⁢B,q⁢Y,13⁢A)	(3⁢C,q⁢Y,13⁢A)	(3⁢D,q⁢Y,13⁢A)	(3⁢E,q⁢Y,13⁢A)
ΔFi24′	5⁢A	45630	1565655	141679395	9800873901	35212760388
	7⁢A	2411370	58501807	6413416191	36548478120420	1838861745084
	7⁢B	13538304	249902172	42649368372	3982990632660	15960483244368
	11⁢A	227135844	4336239492	686421388692	62529358471632	248339382279768
ΣFi23	5⁢A	11466	197431	6290739	169703937	—
	7⁢A	359970	5157763	173434755	109914610680	—
	11⁢A	6737640	64646400	2622896640	98122752000	—
Σ2⋅Fi22:2	5⁢A	2158	24427	392535	3090321	—
	7⁢A	36998	354419	5794191	285717348	—
	11⁢A	149760	1164384	20966400	167731200	—
Σ(3×O8+(3):3):2	5⁢A	1404	17940	158652	687960	0
	7⁢A	25272	372736	3462264	60528384	—
Σ37⋅O7⁢(3)	5⁢A	260	728	2132	258024	210600
	7⁢B	13260	43056	271440	19270680	18195840
Σ(32:2×G2(3)).2	7⁢A	0	1066	3978	3528	0
ΣL2⁢(13):2	7⁢B	—	—	—	78	—

3.3.1 The subcase q=5

Proposition 3.13.

Fi24′ is not (3⁢A,5⁢A,13⁢A)-generated.

Proof.

We compute Δ⁢(Fi24′)=45630. A fixed z∈13⁢A is contained in 6 conjugates of M≅Fi23, in 9 conjugates of A≅2⁢F⁢i22⁢.2, and in 2 conjugates of H≅(3×O8+⁢(3)⁢.3)⁢.2. Also z is contained in 12 conjugates of isomorphism type 37.O7⁢(3) but we shall not make direct use of this fact. These are the only maximal subgroups of Fi24′ that can contribute to Δ⁢(Fi24′) for the triple (3⁢A,5⁢A,13⁢A).

To investigate potential (3⁢A,5⁢A,13⁢A)-generation of Fi24′, we apply the principle of inclusion-exclusion. As usual, we start by computing the contribution from each of its maximal subgroups. From Table 2, we observe that contributions are coming from 2.Fi22 and Fi23.

We first claim that Σ*(2.Fi22)=1846. We easily compute Σ(2.Fi22)=2158 from its character table. The only maximal subgroups of 2.Fi22 that can contribute to Σ(2.Fi22) are the two non-conjugate classes of 2×O7⁢(3). A fixed element z of Fi24′-type 13A is easily shown to lie in exactly one conjugate O, O′ from each class. We claim that Σ⁢(O∩O′)=0. Indeed, the only remaining possibility is that O∩O′ contains a subgroup L≅L4⁢(3). But this would necessarily fuse the two conjugacy classes of 2×L4⁢(3) subgroups in 2.Fi22, a contradiction. Thus Σ⁢(O∩O′)=0, which implies that

Σ*(2.Fi22)=Σ(2.Fi22)-Σ(O)-Σ(O′)=2158-2(156)=1846.

We next claim that Σ*⁢(Fi23)≥4368. First observe that a fixed z of Fi24′-type 13⁢A is contained in three conjugates of F≅2.Fi22 in Fi23. Also z is contained in a unique conjugate of each of H≅O8+⁢(3).S3 and K≅S3×O7⁢(3) in Fi23. Note that these are the only maximal subgroups of Fi23 that can contribute to Δ⁢(Fi24′) for the triple (3⁢A,5⁢A,13⁢A). We compute Σ⁢(Fi23)=11466 from its character table.

Clearly, Δ*⁢(A)=0 as A has abelian quotient and 3,5,13 are pairwise co-prime. However, each of the 9 conjugates of A contains a unique F≅2.Fi22. Let H, Hg be the two conjugates of H that contain z. A glance at the list of maximal subgroups of O8+⁢(3) reveals that there are only two possibilities that can lead to Σ⁢(H∩Hg)>0. Specifically, these are H∩Hg>K where K is a subgroup of either K1≅O7⁢(3) or K2≅36.L4⁢(3). As Σ⁢(K1)=156 and Σ⁢(K2)=260, the worst case scenario is H∩Hg=K2, whereby the total contribution to Δ⁢(Fi24′) is at least

Σ⁢(H)+Σ⁢(Hg)-Σ⁢(K2)=2⁢(1404)-260=2548.

But we now see that

Δ*⁢(Fi24′)≤Δ⁢(Fi24′)-6⁢Σ*⁢(M)-9⁢Σ*⁢(F)-2548=260.

As |CFi24⁢(z)|=468>260, we conclude from Lemma 2.3 that (3⁢A,5⁢A,13⁢A)-generation of Fi24′ is not possible. ∎

Proposition 3.14.

Fi24′ is (3⁢X,5⁢A,13⁢A)-generated for all X∈{B,C,D,E}.

Proof.

The maximal subgroups of Fi24′ of order divisible by 3⋅5⋅13 are

Fi23,2.Fi22:2,(3×O8+(3):3):2,37⋅O7(3).

However, each of Fi23, 2.Fi22:2, (3×O8+(3):3):2 fails to meet the Fi24′-class 3⁢E. Further, a fixed element of Fi24′-type 13⁢A is contained in six conjugates of Fi23, nine conjugates of 2.Fi22:2, two conjugates of (3×O8+(3):3):2 and twelve conjugates of 37⋅O7⁢(3). Hence, from Tables 2 and 6, we obtain

ΔFi24′*⁢(3⁢B,5⁢A,13⁢A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)-2⁢Σ⁢((3×O8+⁢(3)⁢.3)⁢.2)-12⁢Σ⁢(37⋅O7⁢(3))=1565655-6⁢(197431)-9⁢(24427)-2⁢(17940)-12⁢(728)>0,

ΔFi24′*⁢(3⁢C,5⁢A,13⁢A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)-2⁢Σ⁢((3×O8+⁢(3)⁢.3)⁢.2)-12⁢Σ⁢(37⋅O7⁢(3))=141679395-6⁢(6290739)-9⁢(392535)-2⁢(158652)-12⁢(2132)>0,

ΔFi24′*⁢(3⁢D,5⁢A,13⁢A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)-2⁢Σ⁢((3×O8+⁢(3)⁢.3)⁢.2)-12⁢Σ⁢(37⋅O7⁢(3))=9800873901-6⁢(169703937)-9⁢(3090321)-2⁢(687960)-12⁢(258024)>0,

ΔFi24′*⁢(3⁢E,5⁢A,13⁢A)≥Δ⁢(Fi24′)-2⁢Σ⁢((3×O8+⁢(3)⁢.3)⁢.2)-12⁢Σ⁢(37⋅O7⁢(3))=35212760388-2⁢(0)-12⁢(210600)>0.

Thus Fi24′ is (3⁢X,5⁢A,13⁢A)-generated for all X∈{B,C,D,E} as claimed. ∎

3.3.2 The subcase q=7

Proposition 3.15.

Fi24′ is (3⁢A,7⁢A,13⁢A)-generated.

Proof.

We consider the irreducible 3774-dimensional representation of Fi24′ over GF⁢(2). Here it is known that Fi24′ is generated by elements a∈2⁢A, b∈3⁢E with product ab of order 29. Set c=(a⁢b)6⁢b, d=((a⁢b)3⁢b⁢(a⁢b)5⁢b)5, x=c20, y=d6. Then, via GAP, we obtain y∈7⁢A, x⁢y∈13⁢A. Moreover, GAP detects elements in 〈y,x⁢y〉 of respective orders 17, 23, 29. As Fi24′ has no proper subgroups divisible by 17⋅23⋅29, we have Fi24′=〈y,x⁢y〉=〈x,y〉 as claimed. ∎

Proposition 3.16.

Fi24′ is (3⁢X,7⁢A,13⁢A)-generated for all X∈{B,C,D}.

Proof.

The only maximal subgroups that can potentially contribute to the relevant structure constants are Fi23, 2.Fi22:2, (3×O8+(3):3):2, (32:2×G2(3)).2. All structure constants and their respective contributions coming from maximal subgroups are indicated in the computations below (cf. Table 6):

ΔFi24′*⁢(3⁢B,7⁢A,13⁢A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)-2Σ((3×O8+(3).3).2)-Σ((32:2×G2(3)).2)=58501807-6⁢(5157763)-9⁢(351944)-2⁢(372736)-1066>0,

ΔFi24′*⁢(3⁢C,7⁢A,13⁢A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)-2Σ((3×O8+(3).3).2)-Σ((32:2×G2(3)).2)=6413416191-6⁢(173434755)-9⁢(5794191)-2⁢(3462264)-3978>0,

ΔFi24′*⁢(3⁢D,7⁢A,13⁢A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)-2Σ((3×O8+(3).3).2)-Σ((32:2×G2(3)).2)=36548478120420-21⁢(109914610680)-105⁢(285717348)-35⁢(60528384)-70⁢(3528)>0.

Thus generation of each asserted type is established. ∎

Proposition 3.17.

Fi24′ is (3⁢E,7⁢A,13⁢A)-generated.

Proof.

Among all maximal subgroups, only (3×O8+(3):3):2 and (32:2×G2(3)).2 meet all relevant Fi24′-classes. The result now follows since both maximal subgroups have zero structure constant while ΔFi24′⁢(3⁢E,7⁢A,13⁢A)=1838861745084. ∎

Proposition 3.18.

Fi24′ is (3⁢X,7⁢B,13⁢A)-generated for all X∈{A,B,C,E}.

Proof.

The only maximal subgroup meeting each of the Fi24′-classes 3⁢A, 7⁢B, 13⁢A is 37⋅O7⁢(3), which in fact meets each of 3⁢B, 3⁢C, 3⁢D, 3⁢E as well. (However, L2⁢(13):2 also meets the class 3⁢D, which is why we postpone its treatment to Proposition 3.19.)

As a fixed element of type 13⁢A is contained in twelve conjugates of 37⋅O7⁢(3), we conclude from Table 6 that

ΔFi24′*⁢(3⁢X,7⁢B,13⁢A)≥Δ⁢(Fi24′)-12⁢Σ⁢(37⋅O7⁢(3))>0,

which establishes (3⁢X,7⁢B,13⁢A)-generation of Fi24′ for all X∈{A,B,C,E}. ∎

Proposition 3.19.

Fi24′ is (3⁢D,7⁢B,13⁢A)-generated.

Proof.

The only maximal subgroups meeting each of the Fi24′-classes 3⁢A,7⁢B,13⁢A are 37⋅O7⁢(3) and L2⁢(13):2. Also a fixed element of Fi24′-type 13⁢A is contained in twelve conjugates of 37⋅O7⁢(3) and eighteen conjugates of each L2⁢(13):2. Thus, by Table 6, we obtain

ΔFi24′*⁢(3⁢D,7⁢B,13⁢A)≥Δ(Fi24′)-12Σ(37⋅O7(3))-36Σ(L2(13):2)=3982990632660-12⁢(19270680)-36⁢(78)>0,

thus establishing (3⁢D,7⁢B,13⁢A)-generation of Fi24′. ∎

3.3.3 The subcase q=11

Proposition 3.20.

Fi24′ is (3⁢X,11⁢A,13⁢A)-generated, for all X∈{A,B,C,D,E}.

Proof.

Among the maximal subgroups of order divisible by 13, only Fi23 and 2.Fi22:2 meet the Fi24′-class 11⁢A. Thus, by Table 6, we compute

ΔFi24′*(3X,11A,13A)≥Δ(Fi24′)-6Σ(Fi23)-9Σ(2.Fi22:2)>0

for all X∈{A,B,C,D,E} as desired. ∎

3.4 The case r=17

The only maximal subgroups of Fi24′ with order divisible by 17 are Fi23, O10-⁢(2), He:2 (two classes). We provide Table 7 for all relevant data.

Table 7

Structure constants and contributions from maximal subgroups (r=17).

	qY	(3⁢A,q⁢Y,17⁢A)	(3⁢B,q⁢Y,17⁢A)	(3⁢C,q⁢Y,17⁢A)	(3⁢D,q⁢Y,17⁢A)	(3⁢E,q⁢Y,17⁢A)
ΔFi24′	5⁢A	46410	468367	115753680	9126416034	37362361524
	7⁢A	1815549	27991724	5223792249	483511086172800	1876119288720
	7⁢B	12905244	256041420	41858734500	3981264744600	15913801728000
	11⁢A	231581735	3817762301	685242652185	62214852154350	249976129772300
	13⁢A	120356192	2212586686	375504266556	35047054554732	140380759225652
ΣFi23	5⁢A	12920	72845	4820180	168246246	—
	7⁢A	366129	2698342	133887699	252935207280	—
	11⁢A	7080075	55165663	2528788437	98199866376	—
	13⁢A	7620522	64178808	2773913154	110986504164	—
ΣO10-⁢(2)	5⁢A	51	1462	175304	1195236	—
	7⁢A	153	2890	382976	126172200	—
	11⁢A	2890	73780	9748480	64991850	—
ΣHe:2	5⁢A	—	—	2074	—	26690
	7⁢A	—	—	782	—	13498
	7⁢B	—	—	11101	—	171020

3.4.1 The subcase q=5

Proposition 3.21.

Fi24′ is (3⁢C,5⁢A,17⁢A)-generated.

Proof.

All maximal subgroups of order divisible by 17 meet both the Fi24′-classes 3⁢C and 5⁢A. From Tables 2 and 7, we compute

ΔFi24′*⁢(3⁢C,5⁢A,17⁢A)≥ΔFi24′⁢(3⁢C,5⁢A,17⁢A)-ΣFi23⁢(3⁢C,5⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢C,5⁢A,17⁢A)-2⁢ΣHe:2⁢(3⁢C,5⁢A,17⁢A)=115753680-4820180-2⁢(175304)-2⁢(2074)>0.

This establishes that Fi24′ is (3⁢C,5⁢A,17⁢A)-generated as claimed. ∎

Proposition 3.22.

Fi24′ is (3⁢E,5⁢A,17⁢A)-generated.

Proof.

Only the two classes of He:2 meet both 3⁢E and 5⁢A. Thus, from Tables 2 and 7, we obtain

ΔFi24′*⁢(3⁢E,5⁢A,17⁢A)≥ΔFi24′⁢(3⁢E,5⁢A,17⁢A)-2⁢ΣHe:2⁢(3⁢C,5⁢A,17⁢A)=37362361524-2⁢(26690)>0,

proving (3⁢E,5⁢A,17⁢A)-generation as claimed. ∎

Proposition 3.23.

Fi24′ is (3⁢X,5⁢A,17⁢A)-generated for all X∈{A,B,D}.

Proof.

The only maximal subgroups of order divisible by 17 that meet both 5⁢A and 3⁢X for each X∈{A,B,D} are Fi23 and O10-⁢(2). Once more, consulting Tables 2 and 7, we obtain

ΔFi24′*⁢(3⁢A,5⁢A,17⁢A)≥ΔFi24′⁢(3⁢A,5⁢A,17⁢A)-ΣFi23⁢(3⁢A,5⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢C,5⁢A,17⁢A)=46410-12920-2⁢(51)>0,

ΔFi24′*⁢(3⁢B,5⁢A,17⁢A)≥ΔFi24′⁢(3⁢B,5⁢A,17⁢A)-ΣFi23⁢(3⁢B,5⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢B,5⁢A,17⁢A)=468367-72845-2⁢(1462)>0,

ΔFi24′*⁢(3⁢D,5⁢A,17⁢A)≥ΔFi24′⁢(3⁢D,5⁢A,17⁢A)-ΣFi23⁢(3⁢D,5⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢D,5⁢A,17⁢A)=9126416034-168246246-2⁢(1995236)>0,

thus establishing generation of all types claimed. ∎

3.4.2 The subcase q=7

Proposition 3.24.

Fi24′ is (3⁢C,7⁢A,17⁢A)-generated.

Proof.

Each of Fi23, O10-⁢(2), He:2 (two classes) meets the Fi24′-class 3⁢C as well as 7⁢A. From Tables 2 and 7, we obtain

ΔFi24′*⁢(3⁢C,7⁢A,17⁢A)≥ΔFi24′⁢(3⁢C,7⁢A,17⁢A)-ΣFi23⁢(3⁢C,7⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢C,7⁢A,17⁢A)-2⁢ΣHe:2⁢(3⁢C,7⁢A,17⁢A)=5223792249-133887699-2⁢(382976)-2⁢(782)>0,

which establishes that Fi24′ is (3⁢C,7⁢A,17⁢A)-generated as claimed. ∎

Proposition 3.25.

Fi24′ is (3⁢E,7⁢A,17⁢A)-generated.

Proof.

Among all maximal subgroups of Fi24′ divisible by 17, only the two classes of He:2 meet the Fi24′-classes 3⁢E, 7⁢A. We compute (cf. Tables 2 and 7)

ΔFi24′*⁢(3⁢E,7⁢A,17⁢A)≥ΔFi24′⁢(3⁢E,7⁢A,17⁢A)-2⁢ΣHe:2⁢(3⁢E,7⁢A,17⁢A)=1876119288720-2⁢(13498)>0,

from which the result follows. ∎

Proposition 3.26.

Fi24′ is (3⁢X,7⁢A,17⁢A)-generated for all X∈{A,B,D}.

Proof.

Since both classes of He:2 fail to meet 3⁢C, 3⁢E we need only consider the maximal subgroups Fi23 and O10-⁢(2). The relevant computations now follow from Tables 2 and 7:

ΔFi24′*⁢(3⁢A,7⁢A,17⁢A)≥ΔFi24′⁢(3⁢A,7⁢A,17⁢A)-ΣFi23⁢(3⁢A,7⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢A,7⁢A,17⁢A)=1815549-366129-2⁢(153)>0,

ΔFi24′*⁢(3⁢B,7⁢A,17⁢A)≥ΔFi24′⁢(3⁢B,7⁢A,17⁢A)-ΣFi23⁢(3⁢B,7⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢B,7⁢A,17⁢A)=27991724-2698342-2⁢(2890)>0,

ΔFi24′*⁢(3⁢D,7⁢A,17⁢A)≥ΔFi24′⁢(3⁢D,7⁢A,17⁢A)-ΣFi23⁢(3⁢D,7⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢D,7⁢A,17⁢A)=483511086172800-252935207280-2⁢(126172200)>0.

This establishes (3⁢X,7⁢A,17⁢A)-generation of Fi24′ for all X∈{A,B,D}. ∎

Proposition 3.27.

Fi24′ is (3⁢X,7⁢B,17⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

The only maximal subgroups meeting the Fi24′-class 7⁢B are the two classes of He:2. However, both classes of He:2 meet only 3⁢C, 3⁢E. From Tables 2 and 7, we compute

ΔFi24′*⁢(3⁢C,7⁢B,17⁢A)≥ΔFi24′⁢(3⁢C,7⁢B,17⁢A)-2⁢ΣHe:2⁢(3⁢C,7⁢B,17⁢A)=41858734500-2⁢(11101)>0,

ΔFi24′*⁢(3⁢E,7⁢B,17⁢A)≥ΔFi24′⁢(3⁢E,7⁢B,17⁢A)-2⁢ΣHe:2⁢(3⁢E,7⁢B,17⁢A)=15913801728000-2⁢(171020)>0.

As ΔFi24′⁢(3⁢X,7⁢B,17⁢A)>0 for X∈{A,B,D}, the result follows. ∎

3.4.3 The subcase q=11

Proposition 3.28.

Fi24′ is (3⁢X,11⁢A,17⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

The only maximal subgroups of Fi24′ with order divisible by 11⋅17 are Fi23 and O10-⁢(2). Since neither group meets the Fi24′-class 3⁢E, generation of Fi24′ by the triple (3⁢E,11⁢A,17⁢A) is readily established.

For the remaining Fi24′-classes 3⁢A–3⁢D, we have (cf. Tables 2 and 7):

ΔFi24′*⁢(3⁢A,11⁢A,17⁢A)≥ΔFi24′⁢(3⁢A,11⁢A,17⁢A)-ΣFi23⁢(3⁢A,11⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢A,11⁢A,17⁢A)=231581735-7080075-2⁢(2890)>0,

ΔFi24′*⁢(3⁢B,11⁢A,17⁢A)≥ΔFi24′⁢(3⁢B,11⁢A,17⁢A)-ΣFi23⁢(3⁢B,11⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢B,11⁢A,17⁢A)=3817762301-55165663-2⁢(73780)>0,

ΔFi24′*⁢(3⁢C,11⁢A,17⁢A)≥ΔFi24′⁢(3⁢C,11⁢A,17⁢A)-ΣFi23⁢(3⁢C,11⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢C,11⁢A,17⁢A)=685242652185-2528788437-2⁢(9748480)>0,

ΔFi24′*⁢(3⁢D,11⁢A,17⁢A)≥ΔFi24′⁢(3⁢D,11⁢A,17⁢A)-ΣFi23⁢(3⁢D,11⁢A,17⁢A)-2⁢ΣO10-⁢(2)⁢(3⁢D,11⁢A,17⁢A)=62214852154350-98199866376-2⁢(64991850)>0.

Thus (3⁢X,11⁢A,17⁢A) is a generating triple for Fi24′ for every X∈{A,B,C,D,E}. ∎

3.4.4 The subcase q=13

Proposition 3.29.

Fi24′ is (3⁢X,13⁢A,17⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

Only Fi23 has order divisible by 13⋅17. As this group fails to meet the Fi24′-class 3⁢E, we immediately establish (3⁢E,13⁢A,17⁢A)-generation of Fi24′. Verification of (3⁢X,13⁢A,17⁢A)-generation in all remaining cases follows from the computations below (again, with reference to Tables 2 and 7).

ΔFi24′*⁢(3⁢A,13⁢A,17⁢A)≥ΔFi24′⁢(3⁢A,13⁢A,17⁢A)-ΣFi23⁢(3⁢A,13⁢A,17⁢A)=120356192-7620522>0,

ΔFi24′*⁢(3⁢B,13⁢A,17⁢A)≥ΔFi24′⁢(3⁢B,13⁢A,17⁢A)-ΣFi23⁢(3⁢B,13⁢A,17⁢A)=2212586686-64178808>0,

ΔFi24′*⁢(3⁢C,13⁢A,17⁢A)≥ΔFi24′⁢(3⁢C,13⁢A,17⁢A)-ΣFi23⁢(3⁢C,13⁢A,17⁢A)=375504266556-2773913154>0,

ΔFi24′*⁢(3⁢D,13⁢A,17⁢A)≥ΔFi24′⁢(3⁢D,13⁢A,17⁢A)-ΣFi23⁢(3⁢D,13⁢A,17⁢A)=35047054554732-110986504164>0.∎

3.5 The case r=23

The group Fi24′ has two conjugacy classes of elements of order 23, viz. 23⁢A, 23⁢B. However, as these classes are algebraically conjugate, it suffices to restrict attention to the Fi24′-class 23⁢A.

The only maximal subgroups of Fi24′ with order divisible by 23 are Fi23 and 211⋅M24. We further observe (see Table 2) that the Fi24′-classes 3⁢A,3⁢B,3⁢D fail to meet 211⋅M24 while 3⁢E fails to meet Fi23. Thus 3⁢C is the only Fi24′-class for which both groups must be considered as potentially contributing to the relevant structure constant Δ⁢(Fi24′).

Table 8

Structure constants and contributions from maximal subgroups (r=23).

	qY	(3⁢A,q⁢Y,23⁢A)	(3⁢B,q⁢Y,23⁢A)	(3⁢C,q⁢Y,23⁢A)	(3⁢D,q⁢Y,23⁢A)	(3⁢E,q⁢Y,23⁢A)
ΔFi24′	5⁢A	28819	563086	96122865	9038537585	36204088860
	7⁢A	1612944	29403913	4985710299	356571640804680	1862232689916
	7⁢B	13676904	251582694	42694777134	3984950324250	15961650138720
	11⁢A	213253884	3922136640	665668679040	62128719014400	248856672470400
	13⁢A	120235950	2213976780	375640648254	35045491125672	140388376674852
	17⁢A	1224461142	22510484516	5168704488714	482412368377368	1932300135930348
ΣFi23	5⁢A	10005	88366	4102119	169800651	—
	7⁢A	339250	2884407	127079577	187529194440	—
	11⁢A	6756020	56929600	2459358720	98374348800	—
	13⁢A	7620522	64226626	2774682648	110986273818	—
	17⁢A	13073238	109862568	6387671826	254838301956	—
Σ211⋅M24	5⁢A	—	—	27232	—	259072
	7⁢A	—	—	172224	—	1577984
	11⁢A	—	—	683008	—	4898816

3.5.1 The subcase q=5

Proposition 3.30.

Fi24′ is (3⁢X,5⁢A,23⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

Below we provide all relevant computations that establish the desired generations (cf. Tables 2 and 8):

ΔFi24′*⁢(3⁢A,5⁢A,23⁢Z)≥ΔFi24′⁢(3⁢A,5⁢A,23⁢Z)-ΣFi23⁢(3⁢A,5⁢A,23⁢Z)=28819-10005>0,

ΔFi24′*⁢(3⁢B,5⁢A,23⁢Z)≥ΔFi24′⁢(3⁢B,5⁢A,23⁢Z)-ΣFi23⁢(3⁢B,5⁢A,23⁢Z)=563086-88366>0,

ΔFi24′*⁢(3⁢C,5⁢A,23⁢Z)≥ΔFi24′⁢(3⁢C,5⁢A,23⁢Z)-ΣFi23⁢(3⁢C,5⁢A,23⁢Z)-Σ211⋅M24⁢(3⁢C,5⁢A,23⁢Z)=96122865-4102119-27232>0,

ΔFi24′*⁢(3⁢D,5⁢A,23⁢Z)≥ΔFi24′⁢(3⁢D,5⁢A,23⁢Z)-ΣFi23⁢(3⁢D,5⁢A,23⁢Z)=9038537585-169800651>0,

ΔFi24′*⁢(3⁢E,5⁢A,23⁢Z)≥ΔFi24′⁢(3⁢E,5⁢A,23⁢Z)-Σ211⋅M24⁢(3⁢E,5⁢A,23⁢Z)=36204088860-259072>0.∎

3.5.2 The subcase q=7

Proposition 3.31.

Fi24′ is (3⁢X,7⁢Y,23⁢A)-generated for all X∈{A,B,C,D,E} and Y∈{A,B}.

Proof.

We may immediately dispense of the case Y=B since no maximal subgroup of order divisible by 23 meets the Fi24′-class 7⁢B. For the class 7⁢A, we provide below all relevant computations that ensure the desired generations (see Tables 2 and 8):

ΔFi24′*⁢(3⁢A,7⁢A,23⁢A)≥ΔFi24′⁢(3⁢A,7⁢A,23⁢A)-ΣFi23⁢(3⁢A,7⁢A,23⁢A)=1612944-339250>0,

ΔFi24′*⁢(3⁢B,7⁢A,23⁢A)≥ΔFi24′⁢(3⁢B,7⁢A,23⁢A)-ΣFi23⁢(3⁢B,7⁢A,23⁢A)=29403913-2884407>0,

ΔFi24′*⁢(3⁢C,7⁢A,23⁢A)≥ΔFi24′⁢(3⁢C,7⁢A,23⁢A)-ΣFi23⁢(3⁢C,7⁢A,23⁢A)-Σ211⋅M24⁢(3⁢C,7⁢A,23⁢Z)=4985710299-127079577-172224>0,

ΔFi24′*⁢(3⁢D,7⁢A,23⁢A)≥ΔFi24′⁢(3⁢D,7⁢A,23⁢A)-ΣFi23⁢(3⁢D,7⁢A,23⁢A)=356571640804680-187529194440>0,

ΔFi24′*⁢(3⁢E,7⁢A,23⁢A)≥ΔFi24′⁢(3⁢E,7⁢A,23⁢A)-Σ211⋅M24⁢(3⁢E,7⁢A,23⁢A)=1862232689916-1577984>0.∎

3.5.3 The subcase q=11

Proposition 3.32.

Fi24′ is (3⁢X,11⁢A,23⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

Once again, we direct the reader’s attention to Tables 2 and 8. All asserted generations are verified by the following computations:

ΔFi24′*⁢(3⁢A,11⁢A,23⁢A)≥ΔFi24′⁢(3⁢A,11⁢A,23⁢A)-ΣFi23⁢(3⁢A,11⁢A,23⁢A)=213253884-6756020>0,

ΔFi24′*⁢(3⁢B,11⁢A,23⁢A)≥ΔFi24′⁢(3⁢B,11⁢A,23⁢A)-ΣFi23⁢(3⁢B,11⁢A,23⁢A)=3922136640-56929600>0,

ΔFi24′*⁢(3⁢C,11⁢A,23⁢A)≥ΔFi24′⁢(3⁢C,11⁢A,23⁢A)-ΣFi23⁢(3⁢C,11⁢A,23⁢A)-Σ211⋅M24⁢(3⁢C,11⁢A,23⁢Z)=665668679040-2459358720-683008>0,

ΔFi24′*⁢(3⁢D,11⁢A,23⁢A)≥ΔFi24′⁢(3⁢D,11⁢A,23⁢A)-ΣFi23⁢(3⁢D,11⁢A,23⁢A)=62128719014400-98374348800>0,

ΔFi24′*⁢(3⁢E,11⁢A,23⁢A)≥ΔFi24′⁢(3⁢E,11⁢A,23⁢A)-Σ211⋅M24⁢(3⁢E,11⁢A,23⁢A)=248856672470400-4898816>0.∎

3.5.4 The subcase q=13

Proposition 3.33.

Fi24′ is (3⁢X,13⁢A,23⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

Here Fi23 is the only maximal subgroup of Fi24′ that we need to consider. As Fi23 fails to meet the class 3⁢E, and as ΔFi24′⁢(3⁢E,13⁢A,23⁢A)>0, we have that Fi24′ is (3⁢E,13⁢A,23⁢A)-generated. For the remaining cases 3⁢A–3⁢D, we provide the required computations below:

ΔFi24′*⁢(3⁢A,13⁢A,23⁢A)≥ΔFi24′⁢(3⁢A,13⁢A,23⁢A)-ΣFi23⁢(3⁢A,13⁢A,23⁢A)=120235950-7620522>0,

ΔFi24′*⁢(3⁢B,13⁢A,23⁢A)≥ΔFi24′⁢(3⁢B,13⁢A,23⁢A)-ΣFi23⁢(3⁢B,13⁢A,23⁢A)=2213976780-64226626>0,

ΔFi24′*⁢(3⁢C,13⁢A,23⁢A)≥ΔFi24′⁢(3⁢C,13⁢A,23⁢A)-ΣFi23⁢(3⁢C,13⁢A,23⁢A)=375640648254-2774682648>0,

ΔFi24′*⁢(3⁢D,13⁢A,23⁢A)≥ΔFi24′⁢(3⁢D,13⁢A,23⁢A)-ΣFi23⁢(3⁢D,13⁢A,23⁢A)=35045491125672-110986273818>0.∎

3.5.5 The subcase q=17

Proposition 3.34.

Fi24′ is (3⁢X,17⁢A,23⁢A)-generated for all X∈{A,B,C,D,E}.

Proof.

The only maximal subgroup with order divisible by 17⋅23 is Fi23. As already observed, Fi24′ fails to meet the class 3⁢E; however it meets all other classes of elements of order 3. The computations below establish generations of every type claimed:

ΔFi24′*⁢(3⁢A,17⁢A,23⁢A)≥ΔFi24′⁢(3⁢A,17⁢A,23⁢A)-ΣFi23⁢(3⁢A,17⁢A,23⁢A)=1224461142-13073238>0,

ΔFi24′*⁢(3⁢B,17⁢A,23⁢A)≥ΔFi24′⁢(3⁢B,17⁢A,23⁢A)-ΣFi23⁢(3⁢B,17⁢A,23⁢A)=22510484516-109862568>0,

ΔFi24′*⁢(3⁢C,17⁢A,23⁢A)≥ΔFi24′⁢(3⁢C,17⁢A,23⁢A)-ΣFi23⁢(3⁢C,17⁢A,23⁢A)=5168704488714-6387671826>0,

ΔFi24′*⁢(3⁢D,17⁢A,23⁢A)≥ΔFi24′⁢(3⁢D,17⁢A,23⁢A)-ΣFi23⁢(3⁢D,17⁢A,23⁢A)=482412368377368-1254838301956>0.∎

3.6 The case r=29

The only maximal subgroup of order divisible by 29 is 29:14, i.e., the normalizer of a Sylow 29-subgroup in Fi24′. As 3 does not divide this order, we trivially have the following:

Proposition 3.35.

Fi24′ is (3⁢X,q⁢Y,29⁢Z)-generated for all X∈{A,B,C,D,E}, q⁢Y∈{5⁢A,7⁢A,7⁢B,11⁢A,13⁢A,17⁢A,23⁢A,23⁢B}, and Z∈{A,B}.

Proof.

One need only confirm that Δ⁢(Fi24′)>0 for each indicated triple. ∎

The proof of our main result now follows from Propositions 3.6–3.35.

Communicated by Timothy C. Burness

Funding source: King Abdulaziz City for Science and Technology

Award Identifier / Grant number: 13-MAT264-08

Funding statement: This work was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Grant No. 13-MAT264-08.

Acknowledgements

The authors are grateful to an anonymous referee, for catching errors in an earlier draft of the paper and for suggesting ways to greatly improve its readability.

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Received: 2018-04-21

Revised: 2018-10-28

Published Online: 2019-01-20

Published in Print: 2019-05-01

Artikel in diesem Heft

https://doi.org/10.1515/jgth-2018-0104

(3, 𝑞, 𝑟)-generations of Fischer's sporadic group Fi′24

Artikel

Abstract

1 Introduction

Theorem.

2 Preliminaries

Theorem 2.1 (Ganief and Moori [21]).

Lemma 2.2 (Woldar [29]).

Lemma 2.3 (Conder, Wilson and Woldar [10]).

3 (3,q,r)-Generation of Fi24′

3.1 The case r=7

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Proposition 3.6.

Proof.

Proposition 3.7.

Proof.

Proposition 3.8.

Proof.

Proposition 3.9.

Proof.

Proposition 3.10.

Proof.

3.2 The case r=11

3.2.1 The subcase q=5

Proposition 3.11.

Proof.

3.2.2 The subcase q=7

Proposition 3.12.

Proof.

3.3 The case r=13

3.3.1 The subcase q=5

Proposition 3.13.

Proof.

Proposition 3.14.

Proof.

3.3.2 The subcase q=7

Proposition 3.15.

Proof.

Proposition 3.16.

Proof.

Proposition 3.17.

Proof.

Proposition 3.18.

Proof.

Proposition 3.19.

Proof.

3.3.3 The subcase q=11

Proposition 3.20.

Proof.

3.4 The case r=17

3.4.1 The subcase q=5

Proposition 3.21.

Proof.

Proposition 3.22.

Proof.

Proposition 3.23.

Proof.

3.4.2 The subcase q=7

Proposition 3.24.

Proof.

Proposition 3.25.

Proof.

Proposition 3.26.

Proof.

Proposition 3.27.

Proof.

3.4.3 The subcase q=11

Proposition 3.28.

Proof.

3.4.4 The subcase q=13

Proposition 3.29.

Proof.

3.5 The case r=23

3.5.1 The subcase q=5

Proposition 3.30.

Proof.

3.5.2 The subcase q=7

(3, 𝑞, 𝑟)-generations of Fischer's sporadic group Fi′₂₄