A systematic approach to symmetric presentations I: Involutory generators.

R. T. CURTIS School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT

A. M. A. HAMMAS Department of Mathematics, University of Medina, Saudi Arabia

and J. N. BRAY School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT

(Received 8 June 1994; revised 30 September 1994)

Abstract In this paper we conduct a systematic, computerized search for groups generated by small, but highly symmetric, sets of involutions. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. The techniques of symmetric generation developed elsewhere are described afresh, and the results are presented in a convenient tabular form.

1 Introduction

Let G be a homomorphic image of the triangle group (2, 3, m), where m ∉ 3ℤ. Thus G is generated by x, an involution, and y, an element of order 3, whose product is of order m. Alternatively G is generated by t_{0} = x, t_{1} = x^{y} and t_{2} = x^{y^{2}} , (since (yx)^{m} = y^{m}x^{y^{m−1}}..x^{y}x), a highly symmetric set of three involutions. In practice it is very rare for a finite simple group not to possess such a generating set, although verification in a particular case will generally entail some detailed analysis involving structure constants. It is natural to extend this observation to generating sets with designated groups of symmetries, and considerable work has been done in this direction when that group is cyclic or the full symmetric group [2, 13]. In addition, the first author has used other groups of symmetries to construct many of the sporadic simple groups [6, 7]. In this paper, which is based on the second author’s Ph.D. thesis, [10], we establish a systematic approach to the procedure, and present our results in a convenient tabular form.

2 Symmetric generation of a group

Let G be a group and

T = {t_{1}, t_{2}, …, t_{n}} ⊆ G,

then we define

T̄ = {T_{1}, T_{2}, …, T_{n}},

where T_{i} = ⟨t_{i}⟩, the cyclic subgroup generated by t_{i}; we further define N = N_{G}(T̄), the set normalizer in G of T̄. If the following two conditions hold:

(i) G = ⟨T ⟩, and

(ii) N permutes T̄ transitively

then we say that T is a symmetric generating set^{1} for G. In these circumstances we call N the control subgroup. Note that (i) and (ii) imply that G is a homomorphic image of the progenitor

m^{⋆n}:N

a free product of n copies of the cyclic group C_{m} extended by N, where m is the order of t_{i}. Of course, if m = 2 then N will simply act as permutations of the n symmetric generators; but, in the more general case, elements of N will have monomial action. Any additional relation required to define G can be written in the form

π = w(t_{1}, t_{2}, …, t_{n}),

where π ∈ N and w is a word in the symmetric generators.

Notation. In what follows we shall let i stand for the coset Nt_{i}, ij for the coset Nt_{i}t_{j} and so on. We shall also let i stand for the symmetric generator t_{i} when there is no danger of confusion. Thus we write, for instance,

ij ∼ k to mean Nt_{i}t_{j} = Nt_{k}.

Writing ij = k would be the much stronger statement that t_{i}t_{j} = t_{k}. Now since

t_{i}π = πt_{i}^{π},

(or iπ = πi^{π} as we shall more commonly write), the permutations involved in any element of G can be gathered on the left. Thus any element of G can be written as a permutation belonging to N followed by a word in the symmetric generators. Indeed, in the case of the progenitor itself, this representation is unique provided the obvious cancellations are performed. Thus, if NxN is a double coset of N in G, we have:

NxN = NπwN = NwN,

where x = πw ∈ G, with π ∈ N, and w is a word in the t_{i}. We denote this double coset by [w], e.g., [01] denotes the double coset Nt_{0}t_{1}N. The double coset NeN = N is denoted by [⋆], where e is the identity element. Furthermore, in [7] we define

N^{i} = C_{N}(t_{i}); N^{ij} = C_{N}(⟨t_{i}, t_{j}⟩) etc.

1It should be pointed out that the previous definition used in [7] required N to act doubly-transitively on T̄, but we now prefer the more general definition.

We now define the coset stabilizing subgroup

N^{(w)} = {π ∈ N | Nwπ = Nw},

for w a word in the symmetric generators. Clearly N^{w} ⩽ N^{(w)}, and the number of cosets in the double coset [w] = NwN is given by |N|/|N^{(w)}|, since

Nwπ_{1} ≠ Nwπ_{2} ⟺ Nwπ_{1}π_{2}^{−1} ≠ Nw
⟺ π_{1}π_{2}^{−1} ∉ N^{(w)}
⟺ N^{(w)}π_{1} ≠ N^{(w)}π_{2}.

3 The progenitors

In this paper we consider the case when the symmetric generators are involutions. Thus we seek homomorphic images of the progenitor:

2^{⋆n}:N

where N is a transitive permutation group on n letters. We shall limit ourselves to N ≅ S_{3}, A_{4}, S_{4}, A_{5}, L_{2}(5) and L_{3}(2) acting on 3, 4, 4, 5, 6 and 7 points respectively. Here A_{n} and S_{n} denote respectively the alternating and symmetric groups on n letters, and L_{n}(q) denotes the projective special linear group in n dimensions over the field of order q in its permutation action on the 1-dimensional subspaces. Our presentations for these progenitors will, in practice, take a standard form namely

⟨x, y, t | ⟨x, y⟩ ≅ N, [t, N^{0}] = 1 = t^{2}⟩ ≅ 2^{⋆n}:N,

where the set of relations includes a standard presentation for N in terms of the generators x and y, see [5], and N^{0} denotes the point stabilizer in N in its action on n letters—generators for which are given as words in x and y. In Table 1 we give an explicit presentation for each of our progenitors, together with the action of the generators of N on the symmetric generators.

Now, as stated above, every element of the progenitor is uniquely expressible as a permutation of N followed by a word in the symmetric generators (provided cancellation of adjacent repetitions is carried out). Double cosets [w] = NwN are then in 1-to-1 correspondence with the orbits of N on ordered k-tuples. The Cayley graph for the progenitor over N is a tree, since any circuit would imply a non-trivial relation between the symmetric generators. For example the (infinite) Cayley graph of the progenitor in (vi) of Table 1 over N ≅ L_{3}(2) has, for l(w) ⩽ 3, the following diagram:

Table 1: Presentations of the progenitors

(i)  ⟨x, y, t | x^{3} = y^{2} = (xy)^{2} = 1 = t^{2} = [t, y]⟩ ≅ 2^{⋆3}:S_{3}
x ∼ (0, 1, 2), y ∼ (1, 2).
(ii)  ⟨x, y, t | x^{3} = y^{3} = (xy)^{2} = 1 = t^{2} = [t, x]⟩ ≅ 2^{⋆4}:A_{4}
x ∼ (1, 2, 3), y ∼ (0, 1, 2).
(iii)  ⟨x, y, t | x^{4} = y^{2} = (xy)^{3} = 1 = t^{2} = [t, y] = [t^{x}, y]⟩ ≅ 2^{⋆4}:S_{4}
x ∼ (0, 1, 2, 3), y ∼ (2, 3).
(iv)  ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = [t, y] = [t, y^{x^{2}}]⟩ ≅ 2^{⋆5}:A_{5}
x ∼ (0, 1, 2, 3, 4), y ∼ (4, 2, 1).
(v)  ⟨x, y, t | x^{5} = y^{3} = (xy)^{2} = 1 = t^{2} = [t, x] = [t^{yx^{2}}, xy]⟩ ≅ 2^{⋆6}:L_{2}(5)
x ∼ (0, 1, 2, 3, 4), y ∼ (∞, 0, 1)(2, 4, 3).
(vi) ⟨x, y, t | x^{7} = y^{2} = (xy)^{3} = [x, y]^{4} = 1 = t^{2} = [xy, t^{x^{4}}] = [y, t^{x^{3}}]⟩ ≅ 2^{⋆7}:L_{3}(2)
x ∼ (0, 1, 2, 3, 4, 5, 6), y ∼ (2, 6)(4, 5).

[figure: 42; 1; 7; 42; 42; 168; 7; 1; 6; 1; 1; 1; 4; 1; 1; 1; [⋆]; [0]; [01]; [010]; [012]; [013]]

4 Additional relations

We now seek finite homomorphic images of such a progenitor, and must consider what additional relation or relations to factor by. As stated above, this is equivalent to asking how to write permutations of N in terms of the symmetric generators T . Now Lemmas 1 and 1′ of [6] state that

⟨t_{i}, t_{j}⟩ ∩ N ⩽ C_{N}(N^{ij})

and, more generally, that

⟨t_{i_{1}}, t_{i_{2}}, …, t_{i_{m}}⟩ ∩ N ⩽ C_{N}(N^{i_{1}i_{2}...i_{m}})

where N^{ij} stands for the stabilizer in N of the points i and j, and N^{i_{1}i_{2}...i_{m}} stands for the stabilizer in N of i_{1}, i_{2}, …, i_{m}.

Example. As an example let us consider the progenitor (iv) of Table 1, namely 2^{⋆4}:S_{4}, when N^{01} = ⟨(2, 3)⟩ and so:

⟨t_{0}, t_{1}⟩ ∩ N ⩽ C_{N}(N^{01}) = ⟨(0, 1), (2, 3)⟩.

Now for involutory symmetric generators we have ⟨t_{0}, t_{1}⟩ ≅ D_{2k}, a dihedral group of order 2k. If k is even, this group has centre of order 2 generated by (t_{0}t_{1})^{k/2}; if k is odd, then the element t_{0}t_{1} … t_{0} of length k interchanges t_{0} and t_{1} by conjugation. Thus, in the first case, we have the possibility

(t_{0}t_{1})^{k/2} = (2, 3),

and in the second

[(0, 1)t_{0}]^{k} = 1 or [(0, 1)(2, 3)t_{0}]^{k} = 1.

Consider first the group defined by

\frac{2^{⋆4}:S_{4}}{(0, 1) = t_{0}t_{1}t_{0}} ,  (1)

the smallest possibility in the odd case. Since 010 ∼ ⋆ we have [010] = [⋆], and so [01] = [0]. Thus there are just two double cosets: [⋆] = N, and [0] which contains four single cosets. We have the double coset diagram

[figure: 1; 4; 4; 1; 3]

The symmetric generator t_{0} acts on these five single cosets by right multiplication as the transposition interchanging coset ∞ with coset 0. Since (∞, 0)(∞, 1)(∞, 0) = (0, 1) we see that our additional relation is satisfied by these transpositions, and we have a symmetric presentation of S_{5}. The smallest possibility in the even case is

\frac{2^{⋆4}:S_{4}}{(2, 3) = [t_{0}t_{1}]^{2}} .  (2)

We now have 0101 ∼ ⋆ ⇒ 01 ∼ 10, and so the coset stabilizing subgroup N^{(01)} ⩾ ⟨(0, 1), (2, 3)⟩. Thus the double coset [01] contains at most 24/4 = 6 single cosets. Moreover we see that

012 ∼ 0.1212.21 ∼ 0.(0, 3).21 ∼ 321,

and so N^{(012)} ⩾ ⟨(0, 3)(1, 2), (0, 1)⟩ ≅ D_{8}, a transitive subgroup of N. This indicates the double coset diagram

[figure: 1; 4; 6; 3; [⋆]; [0]; [01]; [012]; 4; 3; 2; 1; 2; 4]

As above this gives the action of our symmetric generators as permutations on (1+4+6+3) = 14 points, and we readily verify that they satisfy the additional relation required. Alternatively we may recognise the group we have constructed, in this case the projective general linear group PGL_{2}(7), and seek elements in it satisfying the symmetric presentation. If we let Λ = {0, 1, 2, 3} the points of the Fano plane are represented by ⋆ together with the six unordered pairs of members of Λ, and its lines by the four members of Λ together with the three partitions of Λ into two pairs. If λ = {i, j, k, l} then ij.kl, one of the three partitions of the four letters into two pairs, represents the coset ijk ∼ ijl ∼ kli ∼ klj. The three points on each line are given by

i = {kl, lj, jk} and ij.kl = {⋆, ij, kl}.

Again, it may be verified mechanically, see [11, page 64], that

\frac{2^{⋆4}:S_{4}}{[(0, 1)(2, 3)t_{0}]^{5}} ≅ (3 × L_{2}(11)):2.  (3)

Now in each of the six progenitors listed above, the control subgroup acts doubly transitively on the symmetric generators. Applying Lemma 1 of [6] we see that

(i) N ∩ ⟨t_{0}, t_{1}⟩ is unrestricted,
(ii) N ∩ ⟨t_{0}, t_{1}⟩ is unrestricted,
(iii) N ∩ ⟨t_{0}, t_{1}⟩ ⩽ ⟨(0, 1), (2, 3)⟩,
(iv) N ∩ ⟨t_{0}, t_{1}⟩ ⩽ ⟨(2, 3, 4)⟩,
(v) N ∩ ⟨t_{∞}, t_{0}⟩ ⩽ ⟨(∞, 0)(1, 4), (1, 4)(2, 3)⟩,
(vi) N ∩ ⟨t_{0}, t_{1}⟩ ⩽ ⟨(2, 6)(4, 5), (2, 4)(5, 6)⟩,

where the roman numerals refer to the labelling in Table 1 throughout. Now in our case the symmetric generators are involutions and so ⟨t_{0}, t_{1}⟩ is dihedral; we thus obtain further restrictions:

(i) N ∩ ⟨t_{0}, t_{1}⟩ is a subgroup of N normalized by (0, 1) and so is trivial, is ⟨(0, 1)⟩, or is N itself. But if ⟨t_{0}, t_{1}⟩ ⩾ N then we have ⟨t_{0}, t_{1}⟩ = ⟨t_{0}, N⟩ = G, and so G is dihedral. Thus for ‘interesting’ G we must have N ∩ ⟨t_{0}, t_{1}⟩ ⩽ ⟨(0, 1)⟩. The case (0, 1) ∈ ⟨t_{0}, t_{1}⟩ is equivalent to (0, 1)t_{0} having odd order.

(ii) N ∩ ⟨t_{0}, t_{1}⟩ is a subgroup of N normalized by (0, 1)(2, 3) and so is contained in ⟨(0, 1)(2, 3), (0, 2)(1, 3)⟩ or equals N. But (0, 2)(1, 3) ∈ ⟨t_{0}, t_{1}⟩ ⇒ ⟨t_{0}, t_{1}⟩ = ⟨t_{0}, t_{1}, t_{2}, t_{3}⟩ = G, and so G is dihedral. Thus, for ‘interesting’ G, we may assume that N ∩ ⟨t_{0}, t_{1}⟩ is trivial, or that (0, 1)(2, 3)t_{0} has odd order.

(iii) As above (0, 1) or (0, 1)(2, 3) ∈ ⟨t_{0}, t_{1}⟩ is equivalent to (0, 1)t_{0} or (0, 1)(2, 3)t_{0} respectively having odd order. In this case the dihedral group ⟨t_{0}, t_{1}⟩ has trivial centre, and so does not contain (2, 3). If (2, 3) ∈ ⟨t_{0}, t_{1}⟩ then (2, 3) = (t_{0}t_{1})^{s} for some s. Otherwise N ∩ ⟨t_{0}, t_{1}⟩ is trivial.

(iv) A dihedral group cannot have an element of order three in its centre, and so N ∩ ⟨t_{0}, t_{1}⟩ is trivial.

(v) As in (iii) the four mutually exclusive possibilities are that either (∞, 0)(1, 4)t_{∞} or (∞, 0)(2, 3)t_{∞} has odd order, that (1, 4)(2, 3) = (t_{∞}t_{0})^{s} for some s, or that N ∩ ⟨t_{∞}, t_{0}⟩ is trivial.

(vi) We have N ∩⟨t_{0}, t_{1}⟩ ⩽ Z(⟨t_{0}, t_{1}⟩), which has order 1 or 2. Since N ∩⟨t_{0}, t_{1}⟩ is normalized by the element (0, 1)(2, 5), the only possibilities are that (2, 5)(4, 6) = (t_{0}t_{1})^{s} or that N ∩ ⟨t_{0}, t_{1}⟩ is trivial.

Other useful relations by which to factor the progenitor are those of the form (πu)^{s}, where π ∈ N and u is a word in the symmetric generators. In this context we define the orders of elements of the form πt_{i} to be the first order parameters of the symmetric presentation; similarly the orders of elements of the form πt_{i}t_{j} are the second order parameters. In Table 2 we give, for each progenitor, a list of elements whose orders form a complete set of first order parameters, together with other useful parameters. Note that πt_{i} has the same order as πt_{π(i)}, t_{i}π and π^{−1}t_{i}.

5 Some generic images of 2^{⋆3}:S_{3} and other progenitors

In [5] Coxeter and Moser define

G^{p,q,r} ≡ ⟨u, v, w | u^{2} = v^{2} = (uv)^{2} = 1 = w^{2} = (uw)^{p} = (vw)^{q} = (uvw)^{r}⟩.

Thus, if we set p = 3, v = t_{0}, v^{wu} = t_{1}, v^{w} = t_{2}, then by conjugation u and w have actions u ∼ (1, 2), w ∼ (0, 2) on T = {t_{0}, t_{1}, t_{2}}. Consideration of the presentation shows that

G^{3,q,r} ≅ \frac{2^{⋆3}:S_{3}}{[(0, 1)t_{0}]^{q}, [(0, 1, 2)t_{0}]^{r}} ≅ G^{3,r,q}.  (4)

Table 2: The parameters used for each of our progenitors

(i) 2^{⋆3}:S_{3}  (ii) 2^{⋆4}:A_{4}  (iii) 2^{⋆4}:S_{4}
[(0, 1, 2)t_{0}]^{a}  [(0, 1, 2)t_{0}]^{a}  [(0, 1, 2, 3)t_{0}]^{a}
[(0, 1)t_{0}]^{b}  [(0, 1)(2, 3)t_{0}]^{b}  [(0, 1, 2)t_{0}]^{b}
[(0, 1)t_{0}t_{2}]^{c}  [(0, 1, 2)t_{0}t_{3}]^{c}  [(0, 1)(2, 3)t_{0}]^{c}
[(0, 1, 2)t_{0}t_{1}t_{0}]^{d}  [(0, 1)(2, 3)t_{0}t_{2}]^{d}  [(0, 1)t_{0}]^{d}
[(0, 1)t_{0}t_{2}t_{0}]^{e}  [(0, 1)t_{0}t_{2}]^{e}
(2, 3) = [t_{0}t_{1}]^{s}
(iv) 2^{⋆5}:A_{5}  (v) 2^{⋆6}:L_{2}(5)  (vi) 2^{⋆7}:L_{3}(2)
[(0, 1, 2, 3, 4)t_{0}]^{a}  [(0, 1, 2, 3, 4)t_{0}]^{a}  [(0, 1, 2, 3, 4, 5, 6)t_{0}]^{a}
[(0, 2, 4, 1, 3)t_{0}]^{b}  [(0, 2, 4, 1, 3)t_{0}]^{b}  [(1, 2, 4)(3, 6, 5)t_{1}]^{b}
[(0, 1, 2)t_{0}]^{c}  [(∞, 0, 1)(2, 4, 3)t_{∞}]^{c}  [(1, 2, 4)(3, 6, 5)t_{3}]^{c}
[(0, 1)(2, 3)t_{0}]^{d}  [(∞, 0, 1)(2, 4, 3)t_{2}]^{d}  [(2, 6)(4, 5)t_{2}]^{d}
[(∞, 0)(1, 4)t_{∞}]^{e}  [(1, 3)(2, 4, 6, 5)t_{1}]^{e}
[(∞, 0)(1, 4)t_{1}]^{f}  [(1, 3)(2, 4, 6, 5)t_{2}]^{f}
(∞, 0)(1, 4) = [t_{2}t_{3}]^{s}  [(2, 6)(4, 5)t_{0}t_{2}]^{g}
[(2, 6)(4, 5)t_{1}t_{2}]^{h}
(2, 6)(4, 5) = [t_{1}t_{3}]^{s}

i.e., the parameters q and r may be interchanged. This may alternatively be observed by writing s_{0} = t_{0}(1, 2), s_{1} = t_{1}(2, 0) and s_{2} = t_{2}(0, 1), when we see that S = {s_{0}, s_{1}, s_{2}} is a second symmetric generating set permuted by the original control subgroup N = ⟨u, w⟩ ≅ S_{3}. But (0, 1, 2)s_{0} = (0, 2)t_{0} and (0, 1)s_{0} = (0, 2, 1)t_{0}, and (0, 1)s_{0}s_{2} = (0, 1)(1, 2)t_{0}(0, 1)t_{2} = (0, 2)t_{1}t_{2}. Similarly, (0, 1, 2)s_{0}s_{1}s_{0} has the same order as (1, 2)t_{2}t_{0}t_{2}, and (0, 1)s_{0}s_{2}s_{0} has the same order as (0, 1, 2)t_{1}t_{2}t_{1}. Thus, referring to Table 2, the automorphism of the progenitor mapping t_{i} to s_{i} while fixing N, interchanges the roles of parameters a and b, and parameters d and e, while fixing parameter c. For example, the homomorphic image of 2^{⋆3}:S_{3} defined by parameters (7 8 6 4 8) is isomorphic to that defined by (8 7 6 8 4); although, as in this case, the subgroups generated by corresponding subsets of the symmetric generators need not be isomorphic. These groups have been extensively studied and various results are given in [5, page 96], but the adjunction of our second and third order parameters enables us to obtain more factor groups.

It is natural to ask which linear fractional groups are images of this progenitor. Indeed, if we write

(0, 1, 2) ≡ η ↦ \frac{1}{1−η} :(∞, 0, 1)...,
(1, 2) ≡ η ↦ \frac{1}{η}  :(∞, 0)(1)(−1)...,
t_{0} ≡ η ↦ \frac{−1}{η}  :(∞, 0)(1, −1)...,
t_{1} ≡ η ↦ \frac{η−1}{2η−1}
t_{2} ≡ η ↦ \frac{η−2}{η−1}

where η is in the projective line with p + 1 points, we see that all these elements lie in PSL_{2}(p) for p ≡ 1 mod 4. But

(0, 1, 2)t_{0} ≡ η ↦ η − 1,

was shown in [9] to generate PSL_{2}(p) together with t_{0} above. Thus PSL_{2}(p) is an image of 2^{⋆3}:S_{3} for all p ≡ 1 mod 4 and T = {t_{0}, t_{1}, t_{2}} gives a canonical set of symmetric generators. T is still a symmetric generating set in the case p ≡ 3 mod 4, but it may not possess full S_{3} symmetry within the simple group:

Theorem 5.1. The projective special linear group PSL_{2}(p), for p a prime, is a homomorphic image of 2^{⋆3}:S_{3}, except for p = 3, 7 and 11.

Sketch of proof. Let G ≅ L_{2}(p), where p ⩾ 7. We first note that a copy of S_{3} in G must be contained in a (usually) maximal subgroup of shape D_{p+1} or D_{p−1}; and that there is just one conjugacy class of such subgroups, or two classes interchanged by an outer automorphism. Let S_{3} ≅ N ⩽ G, and let y ∈ N have order 2. Then C_{G}(y) ≅ D_{p+1} or D_{p−1}, and so if

Φ = {t ∈ G | t^{2} = 1, [t, y] = 1, t ∉ ⟨y⟩},

then |Φ| = (p + 1)/2 or (p − 1)/2. We must consider what the possibilities are for H = ⟨N, t⟩ with t ∈ Φ. But the maximal subgroups of PSL_{2}(p) are given in [9], and we see that the only possibilities for proper subgroups H are: D_{12}, when p ≡ ±1 mod 12; S_{4} when p ≡ ±1 mod 8; A_{5} when p ≡ ±1 mod 10. When any of these occurs N is in just one copy of H ≅ D_{12}, and in two copies of H ≅ S_{4} or A_{5}. Thus there are possibly 2 elements of Φ extending N to a copy of D_{12}, 2×2 = 4 elements of Φ extending N to a copy of S_{4}, and 2×2 = 4 elements of Φ extending N to a copy of A_{5}. So if |Φ| > 10 then some element of Φ together with N generates G. i.e. if p ⩾ 23 then PSL_{2}(p) is a homomorphic image of 2^{⋆3}:S_{3}. Consideration of the modularity condition on p for subgroups of the three types to exist in G, shows that for p = 19, 17, 13 there are 6, 4, 4 elements of Φ respectively which together with N generate G. However, for p = 11 we have |Φ| = 6, two elements of which generate, together with N, a copy of D_{12} and four of which, together with N, generate a copy of A_{5}; and for p = 7 we have |Φ| = 4 each element of which, together with N, generates a copy of S_{4}. Of course, p = 5 works with the canonical generators given above, and the cases p = 2 and 3 are easy. □

A more detailed version of this proof is given in [1].

Similar, though easier, arguments show that PSL_{2}(p) is a homomorphic image of 2^{⋆4}:S_{4} if, and only if, p ≡ ±1 mod 24; and an image of 2^{⋆6}:L_{2}(5) if, and only if, p ≡ ±1 mod 20.

6 Tabulated results

The information given in the following Tables is, for the most part, self-explanatory. Table 2 describes the particular parameters used for each of our progenitors, and Table 1 allows the reader to write down an explicit presentation for each of the groups given. Note that, in spite of our requirement that T be a generating set for G, we have allowed some interesting cases where this is not quite true! The notation [n] is used for a soluble group of order n, whereas p^{m} denotes an elementary abelian p-group of rank m. Most importantly, the parameters given in bold-face type are those required to define the group; parameters given as small numerals are the values assumed in the resulting group G. Thus a particular group may occur several times, being defined by different subsets of the parameters.

As a general rule we have attempted to identify for the reader the subgroup of G generated by subsets of the symmetric generators, although space restrictions in the tables have on occasions prevented our doing so. For this reason we would mention that, in the case of the homomorphic image of 2^{⋆6}:L_{2}(5) of shape 6·L_{3}(4):2_{3} we have ⟨t_{∞}, t_{0}, t_{1}⟩ ≅ PGL_{2}(7) and ⟨t_{∞}, t_{0}, t_{2}⟩ ≅ 3·PGL_{2}(9). In the same progenitor, when the homomorphic image is L_{2}(19), it is interesting to note that ⟨t_{∞}, t_{0}, t_{1}⟩ ≅ A_{5}, whereas ⟨t_{∞}, t_{0}, t_{2}⟩ is the whole of G!

Verification of the results given in the tables is carried out as follows: For the smaller groups we use coset enumeration over N to find the order of G, and then construct a faithful permutation representation of lower degree to find its structure in detail. Many of the smaller cases have been constructed by hand, see [1], using the double coset enumeration procedure described in [7]. Several interesting Cayley diagrams are given in [1].

For larger groups we exploit the nesting property of the progenitors; thus, for example,

2^{⋆4}:S_{4} ⩽ 2^{⋆7}:L_{3}(2).

Parameters are chosen so that a subgroup generated by a subset of the symmetric generators may be identified. For example, the subgroup generated by an ‘oval’ ⟨t_{0}, t_{3}, t_{6}, t_{5}⟩ in an image of 2^{⋆7}:L_{3}(2) is itself essentially an image of 2^{⋆4}:S_{4}. The fact that

⟨t_{0}, t_{3}, t_{6}, t_{5}⟩ ≅ L_{2}(23)

is used in the identification of M_{24}. It is worth mentioning that the subgroup ⟨x, yxtxy⟩ of [6] has index 24 in the group, even without factoring by the relator [(1, 3)(2, 4, 6, 5)t_{2}]^{11} i.e., not setting the parameter f = 11. This relator is needed for a presentation, however, as is shown by the parameters of 2^{6}:S_{7} in the same table. In obtaining the Tits group ^{2}F_{4}(2)′ as an image of 2^{⋆4}:S_{4} we introduced the parameter e = 12 to identify L_{3}(3) as the subgroup generated by three symmetric generators. It may well not be needed in the presentation.

The low index subgroups facility on Cayley is used to identify groups with a complicated multiplier and outer automorphism group, such as the image 6·L_{3}(4):2_{3} in the progenitor 2^{⋆6}:L_{2}(5). Having found the order of the group by coset enumeration over N, and established a homomorphic image isomorphic to L_{3}(4):2_{3}, the absence of subgroups of index 3, 4 or 6 shows that the derived subgroup is perfect.

Table 3: Some finite images of the progenitor 2^{⋆3}:S_{3}

Parameters  Order Shape of Shape  Shape
a b c d e  of G  ⟨t_{0}, t_{1}⟩ of ⟨T ⟩  of G
5 5  3  3  3  60  D_{10}  G  A_{5}
7 8  6  4  8  336  D_{8} L_{2}(7)  PGL_{2}(7)
8 7  6  8  4  336  D_{14}  G  PGL_{2}(7)
7 9  9  9  7  504  D_{18}  G  L_{2}(8)
8  10 5  10  8  720  D_{10}  G  PGL_{2}(9)
7 13  7  13  6  1 092  D_{26}  G  L_{2}(13)
10 11  12 4  5  1 320  D_{22}  G PGL_{2}(11)
8 10 15  10  8  2 160  D_{10}  G 3·PGL_{2}(9)
12 7  14  14  6  2 184  D_{14}  G PGL_{2}(13)
9 9  10  10  10  3 420  D_{18}  G  L_{2}(19)
17  17 5  17  17  4 080  D_{34}  G  L_{2}(16)
18 9  16  16 4  4 896  D_{18}  G PGL_{2}(17)
10 12 7  6  10  5 040  D_{12}  G  S_{7}
18  20 5 6  20  6 840  D_{20}  G PGL_{2}(19)
13  13 5  12  12  7 800  D_{26}  G  L_{2}(25)
8 13 12  12  8  11 232  D_{26}  G  L_{3}(3):2
8 11  24  24  23  12 144  D_{22}  G PGL_{2}(23)
7 15  29  15  15  12 180  D_{30}  G  L_{2}(29)
28 13  26 4  13  19 656  D_{26}  G PGL_{2}(27)
28 7 10  30  14  24 360  D_{14}  G PGL_{2}(29)
32 8  32  32 5  29 760  D_{8}  G PGL_{2}(31)
7  22 11  11  21  39 732  D_{22}  G  L_{2}(43)
44 11 6  14  21  79 464  D_{22}  G PGL_{2}(43)
15  29 5  29  29 102 660  D_{58}  G  L_{2}(59)
52  52 9 4  54 148 824  D_{52}  G PGL_{2}(53)
10 19 7  10  11 175 560  D_{38}  G  J_{1}
35  36 5 9  36 178 920  D_{36}  G  L_{2}(71)
8 74 9  74  74 388 944  D_{74}  G PGL_{2}(73)
8 15 10  8  21 483 840  D_{30}  G 2^{2}·L_{3}(4):S_{3}
12  15 7 5  15 604 800  D_{30}  G  HJ = J_{2}
115 19 5 115 115 6 004 380  D_{38}  G  L_{2}(229)

Table 4: Some finite images of the progenitor 2^{⋆4}:A_{4}

Parameters  Index of  Order Shape of  Shape  Shape
a b c d N in G  of G ⟨t_{0}, t_{1}, t_{2}⟩  of ⟨T ⟩  of G
4  6  5  5  10  120  S_{4}  G  S_{5}
7  7  6  7  91  1 092  G  G  L_{2}(13)
8 6  20 10  640  7 680  4^{2}:D_{6}  G  4·2^{4}·S_{5}
8 6 10  5  320  3 840  4^{2}:D_{6}  G  2^{1+4}:S_{5}
8  6 5  5  160  1 920  4^{2}:D_{6}  G  2^{4}:S_{5}
8 6 5  5  160  1 920  4^{2}:D_{6}  G  2^{4}:S_{5}
8  8  7 3  28  336  G  G  PGL_{2}(7)
9 9 6  7  63 504 762 048  L_{2}(8)^{2}  L_{2}(8)^{2} (L_{2}(8) × L_{2}(8)):3
10 6 12 13  1 300  15 600  5^{2}:S_{3}  G  PΣL_{2}(25)
10  8 7  5  6 720  80 640 PGL_{2}(9)  G  2·L_{3}(4):2_{3}
10 12 5  6  420  5 040  G  G  S_{7}
10 12 5 6  420  5 040  G  G  S_{7}
11 5  5  6  55  660  G  G  L_{2}(11)
11  5 5  6  55  660  G  G  L_{2}(11)
18  16 4  9  408  4 896  G  G  PGL_{2}(17)
24  8  21 3  84  1 008 PGL_{2}(7) PGL_{2}(7)  3 × PGL_{2}(7)
33 5  15  6  165  1 980  L_{2}(11)  L_{2}(11)  3 × L_{2}(11)
110  60 5 6 277 200 3 326 400  G  G  L_{2}(11) × S_{7}

Parameters  Order Shape of  Shape  Shape
a b c d e s  of G ⟨t_{0}, t_{1}, t_{2}⟩  of ⟨T ⟩  of G
10 8 6  6  8 —  7 680  2^{2}·S_{4}  2^{1+4}:S_{5}  2^{1+4+1}:S_{5}
10 4  6  6  4 —  240  S_{4}  S_{5}  2 × S_{5}
5  4  6 3  4 —  120  S_{4}  G  S_{5}
5  4  6  3  4 —  120  S_{4}  G  S_{5}
6  24  8  8  14 —  2 016 PGL_{2}(7) PGL_{2}(7) S_{3} × PGL_{2}(7)
24 8  8  8 7 —  43 008 PGL_{2}(7)  G 2{1+6 \atop +}:PGL_{2}(7)
6  8  8  8  7 2  336  G  G  PGL_{2}(7)
8  12  12  12  13 3  11 232  G  G  L_{3}(3):2
8 12  12  12  13 3  11 232  G  G  L_{3}(3):2
8 12  20  20  31 5  744 000  G  G  L_{3}(5):2
8 12  28  28  48 7 11 261 376  G  G  SL_{3}(7):2
11  11  12  12  11 3  6 072  G  G  L_{2}(23)
11 11  12  12  11 3  6 072  G  G  L_{2}(23)
12 10  12  12  21 3  15 120  G  G  3·S_{7}
12  10  12  12 7 3  5 040  G  G  S_{7}
12  33 5  10  12 —  3 960  L_{2}(11)  L_{2}(11) (3 × L_{2}(11)):2
12 11 5  10  12 —  1 320  L_{2}(11)  L_{2}(11)  PGL_{2}(11)
14 7  7  14  12 —  2 184  L_{2}(13)  L_{2}(13)  PGL_{2}(13)
30  10  10 5  6 —  14 400  2 × A_{5}  G 2·(A_{5} × A_{5}):2
15  10  10 5  6 —  7 200  2 × A_{5}  G  (A_{5} × A_{5}):2
13  10  6 6  10 —  15 600  5^{2}:S_{3}  G  PΣL_{2}(25)
7  30  30  15  30 —  24 360  G  G  PGL_{2}(29)
25 8  14 7  6 —  117 600 PGL_{2}(7)  G  PΣL_{2}(49)
82 8  10 10 10 — 1 062 720 PGL_{2}(9) L_{2}(81):2_{2} 2 × L_{2}(81):2_{2}
10 10  8  8  30 —  483 840 3·PGL_{2}(9) 6·L_{3}(4):2_{3} 6·L_{3}(4):2_{3} × 2
10  10  8 8 5 —  80 640 PGL_{2}(9)  G  2·L_{3}(4):2_{3}
20 10  8  8 5 —  161 280 PGL_{2}(9)  G  2^{2}·L_{3}(4):2_{3}
9 12 10  10  14 —  362 880  S_{7}  G  2 × A_{9}
9 12  10  10 7 —  181 440  S_{7}  G  A_{9}
10 11  10 10  12 —  190 080  L_{2}(11)  M_{12}  M_{12}:2
10 13  8 8 12 — 35 942 400  L_{3}(3)  ^{2}F_{4}(2)′  2 ×^{2}F_{4}(2)′

Parameters  Index of  Order Shape of  Shape of  Shape
a b c d N in G  of G ⟨t_{0}, t_{1}, t_{2}⟩ ⟨t_{0}, t_{1}, t_{2}, t_{3}⟩  of G
6  6  4  6  12  720  S_{4}  S_{5}  S_{6}
6  6 4  6  12  720  S_{4}  S_{5}  S_{6}
7 19  15 10  2 926  175 560  G  G  J_{1}
8 8  14  8  1 960  117 600 2 × L_{2}(7)  G PΣL_{2}(49)
8  8  14 8  1 960  117 600 2 × L_{2}(7)  G PΣL_{2}(49)
12 12 8 8 333 312 19 998 720 PGL_{2}(7) 2{1+6 \atop +}:PGL_{2}(7)  L_{5}(2):2
13 13 10 6  78 000 4 680 000  5^{2}:S_{3}  PΣL_{2}(25)  S_{4}(5)

Table 7: Some finite images of the progenitor 2^{⋆6}:L_{2}(5)

Parameters  Order  Shape of Shape of  Shape
a b c d e f s  of G ⟨t_{∞}, t_{0}, t_{1}, t_{2}⟩ ⟨T \ {t_{4}}⟩  of G
4  10 10 12 12 12  3  1 320  G  G PGL_{2}(11)
8  8 10 10  8  8 2  2 160  G  G 3·PGL_{2}(9)
9  9 5  9 10  5 —  3 420  G  G  L_{2}(19)
9  9  5  9 10 5 —  3 420  G  G  L_{2}(19)
6 6 12 12  6  6 — 15 120  (3 × A_{5}):2  3·S_{6}  3·S_{7}
7  21  7 41 20 20  5 34 440  G  G  L_{2}(41)
8  8 8  8  6 6 — 190 080  G  G  2 × M_{12}
6 10  8 10  8  8 — 241 920  G  G 6·L_{3}(4):2_{3}

Table 8: Some finite images of the progenitor 2^{⋆7}:L_{3}(2)

Parameters  Order Shape of  Shape of Shape
a b c d e f g h s  of G ⟨t_{0}, t_{1}, t_{3}⟩ ⟨t_{0}, t_{3}, t_{6}, t_{5}⟩  of G
12  6  8  8  8  6  8  7 2  12 096  [2^{5}]  PGL_{2}(7) U_{3}(3):2
12  6  8  8  8 6  8  7  2  12 096  [2^{5}]  PGL_{2}(7) U_{3}(3):2
6  6 10 12  6  12 12 21  3  15 120  6^{2}:2  G  3·S_{7}
6  6 10 12  6  12 4  7  3  5 040 D_{12} × 2  G  S_{7}
8 12 12  6 12  10  4  4 —  40 320  S_{4}  S_{5}  S_{8}
12 12 10 12  6  12 4 7 3  322 560  S_{4} × 2^{2}  S_{7}  2^{6}:S_{7}
12 12 10 12 6  12  4 7  3  322 560  S_{4} × 2^{2}  S_{7}  2^{6}:S_{7}
10 12 8 6 12  10 12  8 —  5 322 240  3:S_{4}  2^{1+4}:S_{5} 6·M_{22}:2
10 12 8 6 12  10 4  8 —  1 774 080  S_{4}  2^{1+4}:S_{5} 2·M_{22}:2
10 12 8  6 12 10 4  8 —  1 774 080  S_{4}  2^{1+4}:S_{5} 2·M_{22}:2
15 12 11 12 6 11 4 11  3 244 823 040  S_{4} × 2^{2}  L_{2}(23)  M_{24}
15 12 11 12  6 11 4 11 3 244 823 040  S_{4} × 2^{2}  L_{2}(23)  M_{24}

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