A PERFECT GROUP OF 8,315,553,613,086,720,000 AND THE SPORADIC SIMPLE GROUPS BY J. H. CONWAY

UNIVERSITY OF CAMBRIDGE, ENGLAND Communicated by Saunders Mac Lane, August 15, 1968 The group of the title is the group (called * 0) consisting of those operations of the 04(Q), which preserve the remarkable A discovered by John Leech1' 2 in connection with close packing of spheres. The group has a center of order 2, the quotient being a -1 of order 22139547211.13.23, and two further new simple groups *2 and *3 of orders 21836537.11.23 and 2103753- 7.11.23 appear as stabilizers of 1-dimensional sublattices of A. Many other interesting insoluble groups, in particular the Mathieu groups, and the simple groups recently discovered by J. McLaughlin3 and by D. G. Higman and C. C. Sims,4 also appear as stabilizers of sublattices of small dimension in A. Professor J. G. Thompson has also discovered simple groups of the same orders as the Hall-Janko and Suzuki sporadic simple groups among the of *0. The geometry of A illuminates many relationships between these various simple groups, as well as, for instance, providing simple proofs of the existence of the McLaughlin and Higman-Sims groups, and of the identity between the latter and the doubly transitive group of degree 176 considered by G. Higman (see ref. 5). Let Q be the set of cardinal 24 consisting of the symbol o and the residue classes modulo 23, and let Q be the subset consisting of 0 and the quadratic residues. The power set P(Q) becomes a vector space over GF(2) when we define A + B as the (A\B) U (B\A), and it is remarkable that the set Q and its images under PSL2(23) (which acts on Q) span a 12-dimen- sional subspace e consisting of 4096 = 1 + 759 + 2576 + 759 + 1 sets of cardi- nals 0,8,12,16,24, respectively, called e-sets. The e3-sets of cardinal 8, called octads, form a S(5,8,24) (see ref. 6), and so the set of all permuta- tions of Q preserving e is the M24. The set [C,m], for C E e and m an , is the set of all integer points x C R24 whose coordinates xj(i C 0) satisfy 2:xi = 4m and also xi m+2 (mod 4) according as inc, and the A is the union of the sets [C,m] over all C E e and all m. A is a lattice, since the vector sum of [C,m] and [D,n] is [C + D, m + n]. There is a Mathieu group M24 of permutation matrices acting on A, and also an elementary group E of order 212 consisting of all matrices ec(C Ee), where 6c = diag(X1), and Xi = 4 1 according as i0C. The group N24 = EM24 is a split extension of E by M24. Now let K be a set of six disjoint 4-element subsets of Q with the property that the union of any two members of K is an octad, and let T C K. Then if vj is the ith coordinate vector, and fl the operation taking vf to vi - 1/2(Vi + v; + Vk + vl) whenever i E {ij,kl} £ K, then the operation tT = 1KET is an automorphism of A, and shows that N24 is a proper of the group *0 of all orthogonal operations preserving A. In fact it can be shown that N24 is a maximal proper subgroup of *0, which is therefore generated by N24, together with any AT, and so by the operations: 398 Downloaded by guest on September 30, 2021 VOL. 61, 1968 MA THEMA TICS: J. H. CONWAY 399

a: vi Vi+l A Vi V2 i y: Vi -10 V-/i visgi(i X Q) (rj(i X7 Q) 5:vi E = EQ:.V - and t = rT, tri3/9(i E Q) t-vi(i E Q), where T = {0,3, co ,15 } is the fixed set of 6, the appropriate set K being {I0,3,co,15},{18,8,14,20},{4,16,17,10},{2,13,11,7},{6,9,19,5},{1,12,22,21}}. These operations have been chosen so as to satisfy many simple relations-in particular the subsets { ay },{ a4,y,5 },{ a43vy,5,e} generate the groups PSL2(23), M24, N24, respectively. The of *0 are most naturally considered in connection with the * o of all isometries of A, obtained by adding the translations of A to *0. We write *L for the pointwise stabilizer of a sublattice L of A in * , saying that L has type a if it is 1-dimensional and defined by two points at distance 4\/a, type abc if it is 2-dimensional defined by a triangle of sides 4V/a,4x/b,4x/c, and using a similar notation in higher dimensions. In the small cases the type symbol determines the class under 0 , but when it does not, we refine the symbol, using abC for a vector of length 4-\a which is the sum of two vectors of lengths 4x/b,4V\c. Thus (for instance) *222 is a simple group of the same order as PSU6(2), *322 is McLaughlin's simple group, *332 the Higman-Sims group, and *432,*542, -63333 the Mathieu groups M23,M22,M12, while *532 is the simple group PSU4(3) extended by an involution, and *333 an elementary group of order 35 extended by Mln. In considering the 1-dimensional lattices, we find *2 and *3, the new simple groups already mentioned; *4 = N23 is an elementary group of order 211 extended by M23, and * 1042(= N22) an elementary group of order 210 extended by M22, while .5, 622, 9331 * 1033,* 1052 are the groups *322, *222, .333, *332, *542 (respectively) extended by involutions, and we have the equalities *632 = M24, *942 = M23, -7 = *332, *832 = *322, and *822 = *2. When we add that *842 is an extra-special group of order 29 extended by A8, we have completed the de- scription for all lattices of type a < 10. The required proofs of transitivity reduce in every case to simple numerical and geometrical computations, which are also used to determine the index of * L in M when the lattice M contains L. One particular calculation determines the order of -0, since it is easily seen that the stabilizer of a certain vector of length 4V\4 is the same in *0 as it is in N24. Several insoluble groups appear among the centralizers of various subgroups of small order. The most notable case is the centralizer of a certain element of order 3, which is a of order 6 extended by a group S. The group S is a simple group of the same order as Suzuki's recently discovered simple group, and has simple subgroups of the same orders as the Hall-Janko group and the simple group G2(4). The simplicity of * 1, *2, *3 was first established by Thompson, using only the orders and the existence of the 24-dimensional rational representation of *0; several easy geometric proofs are now available. The permutation repre- Downloaded by guest on September 30, 2021 400 MATHEMATICS: J. H. CONWAY PROC. N. A. S.

sentation on the cosets of N24 has degree 8,292,375, and the stabilizer of a point has orbits of sizes 1, 3542, 48576, 1457280, 2637824, 4145152, and explicit repre- sentatives are known for the corresponding double cosets. It follows from some computations of M. J. T. Guy that -0 has strictly more than 150 conjugacy classes. I should like above all to thank John Leech for discovering his Lattice and so freely disseminating information about it, John McKay for bringing it to my notice, and Pro- fessor John Thompson for stimulating conversations that began on the day the group was discovered and have not yet ceased. 1 Leech, J., "Some sphere packing in higher space," Can. J. Math., 16, 657-682 (1964). Leech, J., "Notes on sphere packings," Can. J. Math., 19, 251-267 (1967). 3 McLaughlin, J., "A simple group of order 898,128,000," unpublished. 4 Higman, D. G., and C. C. Sims, "A simple group of order 44,352,000," unpublished. Sims, C. C., "On the isomorphism between two groups of order 44,352,000," unpublished. 6 Todd, J. A., "A representation of the Mathieu group M24 as a collineation group," Ann. Mat. (IV), 71, 199-238 (1966). Downloaded by guest on September 30, 2021