We give a brief introduction to sphere-packing in n-dimensions and, in particular, to lattice packings. The notions of density and kissing number are explained. The 24-dimensional Leech lattice Λ is defined following Conway’s approach in his Three lectures on exceptional groups, see Conway (1971), with vectors in the lattice invariably displayed as they appear in the MOG. We explore the factor space Λ/2Λ, a 24-dimensional vector space over ${\mathbb{Z}}_2$, and show that its non-zero elements may be taken to be vectors of type 2 and 3 together with sets of 24 mutually orthogonal vectors of type 4 (and their negatives). These last elements are known as frames of reference or crosses; the six orbits on crosses under permutations of M24 and sign changes on $\mathcal{C}$-sets are described explicitly in the text, as are the orbits on vectors of types 2, 3 and 4. Finally, we explain how Λ can be defined in terms of a single Lorentz-type vector with 25 space-like coordinates and one time-like coordinate.

The binary Golay code $\mathcal{C}$ is defined as the 12-dimensional vector space over ${\mathbb{Z}}_2$ spanned by the 759 octads interpreted as vectors with eight 1s and 16 0s. The MOG is constructed by considering two 3-dimensional spaces over ${\mathbb{Z}}_2$, the Point space and the Line space, whose codewords are of length 8, and gluing three copies together in such a way as to obtain a 12-dimensional subspace of the 24-dimensional space P(Ω), consisting of all subsets of Ω. The minimal weight codewords in this 24-dimensional space are shown to have weight 8 and to total 759. The construction thus proves that a Steiner system S(5, 8, 24) exists, and provides a unique label for each codeword in the binary Golay code. We exhibit a natural isomorphism between the 24-dimensional space P(Ω) factored by $\mathcal{C}$ and the dual space ${\mathcal{C}}^{\star }$, and identify its elements as 24 monads, 276 duads, 2024 triads and $\left(\genfrac{}{}{0pt}{}{24}{4}\right)/6=1771$ sextets; this last division by 6 occurs because two tetrads 4 whose union is an octad are congruent modulo $\mathcal{C}$.

A sextet is a partition of the 24 points of Ω into six tetrads such that the union of any two of them is an octad. We find all ways in which two sextets can intersect one another and use this knowledge to force any Steiner system S(5, 8, 24) to assume the form of the one given by the MOG. In so doing we show that if a Steiner system S(5, 8, 24) exists then the order of its group of automorphisms is 244, 823, 040 and that it acts quintuply transitively on the 24 points. That the MOG does define an S(5, 8, 24) was proved in Chapter 4.

What is the minimal test to decide whether a permutation π ∈ S24 lies in our preferred copy of M24? The space $\mathcal{C}$ is 12 dimensional and so if we choose a basis of 12 codewords of $\mathcal{C}$, apply π to each codeword in the basis and verify that the image is also in $\mathcal{C}$ then π ∈ M24. The 12-dimensional subspace $\mathcal{C}$ is self-orthogonal with respect to the usual inner product, and so $\mathcal{C}={\mathcal{C}}^{\perp }$. Thus a vector is in $\mathcal{C}$ if, and only if, it is orthogonal to every codeword in a basis of $\mathcal{C}$. Now one codeword in our basis may be chosen to be the all 1s vector that is clearly fixed by any permutation; the other 11 can be chosen to be octads. In this chapter we show that we can do much better than this. In fact we show that we can choose 8 octads that are contained in one, and only one, copy of $\mathcal{C}$, but that any set of 7 octads is contained in no copy of $\mathcal{C}$ or in more than one. To this set of 8 octads we add a further 3 to form a basis together with the all 1s codeword. We now have a minimal test for membership of M24: apply π to each of the 8 octads; if the image in each case intersects each of the 11 octads in the basis evenly, then π is in M24, otherwise it is not. When working with M24 we often require an element possessing certain properties. In this chapter we show how to construct elements of shape 18.28, 212 and 16.36. We also reproduce a diagram due to Todd and Conway showing the orbits of M24 on the subsets of Ω.

In his paper Todd (1966), J. A. Todd produced a list of eight maximal subgroups of M24, but he did not claim that the list was complete. Some years later McKay and Wales found a subgroup isomorphic to the linear group L2(7) that they were able to show lay in no copy of a group in Todd’s list. In this chapter we give a short combinatorial proof that the resulting list of nine subgroups is complete; we describe each of the nine in some detail, giving a canonical version of each of them as it appears in the MOG. We also describe two further non-maximal subgroups that are of interest in their own right.

This chapter is devoted to the smaller Mathieu group M12 that is the automorphism group of a Steiner system S(5, 6, 12). It possesses an outer automorphism group of order 2 and a group of shape M12 : 2 is a maximal subgroup of M12, the duum group described in Chapter 8. We introduce a device known as the Kitten, as it does for M12 what the MOG does for M24. Three copies of the 3 × 3 tic-tac-toe board are glued together to form a triangle in which the 132 hexads of the S(5, 6, 12) are readily recognized. The canonical embedding of M12 : 2 in M24 is described in detail. The symmetric group S6 is exceptional in that it possesses an outer automorphism; in this chapter we exhibit the isomorphism

${\text{S}}_6:2\cong \text{P}\text{Γ}{\text{L}}_2(9)\cong {\text{A}}_6{.2}^2$

In this chapter we describe how M24 can be generated by seven involutions that are normalized as a set by a subgroup of M24 isomorphic to the linear group L3(2). These seven elements are described combinatorially as acting on a conjugacy class of 7-cycles in L3(2). They are also described as acting on the 24 heptagonal faces of the Klein map κ, whose 84 edges fall into seven blocks of imprimitivity of size 12 under the action of L3(2); thus there is one permutation for each block. So, remarkably, generators for M24 may be read directly off κ. Analogously we describe how M12 can be generated by five elements of order 3 that are normalized within M12 by a subgroup isomorphic to the alternating group A5. These five elements are described combinatorially as acting on a conjugacy class of 5-cycles in A5. They are also described as acting on the 12 faces of the regular dodecahedron whose 20 vertices fall into five blocks of imprimitivity of size 4 (each of which forms a regular tetrahedron); thus there is one generator for each block. So generators for M12 can be read directly off the faces of the dodecahedron. Galois showed that the only simple groups L2(p) that can act faithfully with degree p are for p = 5, 7 and 11. These constructions for M12 and M24 owe their existence to the cases p = 5 and p = 7.

Properties of the Steiner system S(5, 8, 24) are given including the important Todd triangle that reveals the manner in which the 759 8-element subsets of the system (the octads) intersect one another. The Mathieu group M24 is the group of all permutations of the 24 letters Ω that preserve these 759 octads. We introduce the Miracle Octad Generator or MOG, a device in which the 759 octads are easily recognized, and explain where it comes from. A mnemonic for recovering the standard MOG labelling is given, along with examples of octad finding: that is to say, identifying the unique octad containing any given five points of the 24.

In the final chapter all the ideas of the book come together to produce the chain of subgroups of the Conway simple group Co1 that was previously referred to as the Suzuki chain. Since this construction emphatically reveals that the chain includes Co1 itself, we prefer to call it the Thompson chain as it was John Thompson who first noted that, with one exception, the normalizers of the groups in the chain are maximal in Co1. In a complete graph on n vertices we let the directed edge from vertex r to vertex s correspond to trs, an element of order 7 in some group where $t_{sr}=t_{rs}^{-1}$. We thus obtain a progenitor of shape

$7^{\star \left(\genfrac{}{}{0pt}{}{n}{2}\right)}:{\mathrm{S}}_n$

$〈t_{12},t_{23},t_{31}〉$

We describe the method of symmetric generation of groups that has been used to produce pleasing, well-motivated constructions of many of the sporadic simple groups. A progenitor 2*n : N is defined to be a free product of n copies of the cyclic group of order 2, extended by a permutation group N of degree n that permutes these n involutory symmetric generators by conjugation. A useful lemma is introduced that leads to suitable relations by which to factor the infinite progenitor in order to produce a finite image that contains n distinct involutions permuted by an isomorphic copy of N. These ideas are employed using $N\cong {\text{M}}_{24}$.

In the first case, we have $n=\left(\begin{array}{c}24\\ 4\end{array}\right)$ and the symmetric generators correspond 4 to tetrads of points of Ω. Factoring this by a single short relation results in the Conway group ·O and the symmetric generators spontaneously reveal themselves to be the negatives of the Conway elements ξT.

In the second case, we have n = 3795 and the symmetric generators correspond to the trios. The Lemma leads us to two simple relations and factoring by them results in the largest Janko group J4. In fact one of the relations is essentially redundant and John Bray has shown that omitting it leads to a group J4 × 2. In order to prove these assertions in the two cases, we first use known representations to verify that the group claimed is indeed an image of the progenitor and satisfies the additional relations. We then use the double coset enumerator of John Bray and the author, written in Magma, to show that the image has the right order. Unfortunately in both these cases the group $N\cong {\text{M}}_{24}$ is too small to enumerate double cosets of form NwN and so we use theoretical arguments to identify subgroups $H\cong 2^{12}:{\text{M}}_{24}$ and $H\cong 2^{11}:{\text{M}}_{24}$, respectively, and enumerate double cosets of form HwN.

The hexacode is a 3-dimensional, length 6 code over GF4, the Galois field of order 4, whose codewords are readily remembered. Each of these codewords represents 26 codewords of $\mathcal{C}$ and thus we obtain the 43.26 = 212 codewords of $\mathcal{C}$. Each element of GF4 = {0, 1, ω, ω̄} is given an odd and an even interpretation as a 4-dimensional column vector (corresponding to the columns of the MOG) or its complement: Thus a hexacodeword [1, 0, 0, 1, ω, ω̄] would have a 4-vector corresponding to 1 in the first column of the MOG, 0 in the second, 0 in the third and so on. All entries must be even or all entries odd and the first five columns may be complemented arbitrarily; the sixth column must then be complemented or not so that in the even interpretation the number of entries in the top row is even, and in the odd interpretation it is odd. Once the reader has memorized the hexacodewords, he or she can work with the MOG without the need of a physical copy of Figure 3.2.

The most combinatorially interesting maximal subgroups of M24 are the stabilizers of an octad, a duum, a sextet and a trio. In this chapter we investigate the way in which the stabilizer of one of these objects acts on the others. This involves some basic but fascinating character theory; the approach given here is intended to be self-contained. For each of the four types of object we draw a graph in which each member is joined to members of the shortest orbit of its stabilizer. Thus in the octad graph we join two octads if they are disjoint; we join two dua if they cut one another 8.4/4.8; we join two sextets if the tetrads of one cut the tetrads of the other (22.04)6; and we join two trios if they have an octad in common. A diagram of each of these four graphs is included as is the way in which these graphs decompose under the action of one of the other stabilizers. Each of these graphs is, of course, preserved by M24.

The Conway group ·O is defined to be the group of all symmetries of Λ fixing the origin. The manner in which Λ has been constructed ensures that it is preserved by all permutations of M24 and all sign changes on a $\mathcal{C}$; we now introduce the Conway element ξT that is not in the aforementioned group of shape 212 : M24 and which we can show preserves Λ. We use ξT to prove transitivity on type 2 vectors, type 3 vectors and crosses, and are then able to work out the order of ·O. This construction is now repeated using the algebra package Magma and a full explanation of the computation is included. An important Remark explains how every element of ·O corresponds in a certain sense to one of the crosses. The stabilizers of a type 2 vector and of a type 3 vector are the Conway groups Co2 and Co3, respectively. Each of these sporadic groups is described in some detail, both manually and computationally using Magma. The Classification of Finite Simple Groups (CFSG) states that any finite simple group is either a member of one of the known infinite families or it is one of 26 sporadic groups. We give a brief description of these groups and how they were discovered, and refer the reader to more complete expositions. We conclude by mentioning the remarkable observation made by John McKay that relates the modular function j to the degrees of irreducible representations of the Monster group M. Conway referred to this tantalizing connection with number theory as Monstrous Moonshine.