Sporadic group

Five of the sporadic groups were discovered by Émile Mathieu in the 1860s and the other twenty-one were found between 1965 (J1) and 1975 (J4). Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is: relations between the 26 sporadic groups. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between.The generations of Robert Griess: 1st, 2nd, 3rd, Pariah • Mathieu groups M11, M12, M22, M23, M24 • Janko groups J1, J2 or HJ, J3 or HJM, J4 • Conway groups Co1, Co2, Co3 • Fischer groups Fi22, Fi23, Fi24′ or F3+ • Higman-Sims group HS • McLaughlin group McL • Held group He or F7+ or F7 • Rudvalis group Ru • Suzuki group Suz or F3− • O'Nan group ''O'N'' (ON) • Harada-Norton group HN or F5+ or F5 • Lyons group Ly • Thompson group Th or F3|3 or F3 • Baby Monster group B or F2+ or F2 • Fischer-Griess Monster group M or F1 Various constructions for these groups were first compiled in , including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms. These are also listed online at , updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in . A further exception in the classification of finite simple groups is the Tits group , which is sometimes considered of Lie type or sporadic — it is almost but not strictly a group of Lie type — which is why in some sources the number of sporadic groups is given as 27, instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both. The Tits group is the of the infinite family of commutator groups ; thus in a strict sense not sporadic, nor of Lie type. For these finite simple groups coincide with the groups of Lie type also known as Ree groups of type 2F4. The earliest use of the term sporadic group may be where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.) The diagram is based on . It does not show the numerous non-sporadic simple subquotients of the sporadic groups. == Organization ==

Happy Family Of the 26 sporadic groups, 20 can be seen inside the monster group as subgroups or quotients of subgroups (sections). These twenty have been called the happy family by Robert Griess, and can be organized into three generations. First generation (5 groups): the Mathieu groups Mn for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a permutation group on 24 points. Second generation (7 groups): the Leech lattice All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice: • Co1 is the quotient of the automorphism group by its center {±1} • Co2 is the stabilizer of a type 2 (i.e., length 2) vector • Co3 is the stabilizer of a type 3 (i.e., length ) vector • Suz is the group of automorphisms preserving a complex structure (modulo its center) • McL is the stabilizer of a type 2-2-3 triangle • HS is the stabilizer of a type 2-3-3 triangle • J2 is the group of automorphisms preserving a quaternionic structure (modulo its center). Third generation (8 groups): other subgroups of the Monster Consists of subgroups which are closely related to the Monster group M: • B or F2 has a double cover which is the centralizer of an element of order 2 in M • Fi24′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A") • Fi23 is a subgroup of Fi24′ • Fi22 has a double cover which is a subgroup of Fi23 • The product of Th = F3 and a group of order 3 is the centralizer of an element of order 3 in M (in conjugacy class "3C") • The product of HN = F5 and a group of order 5 is the centralizer of an element of order 5 in M • The product of He = F7 and a group of order 7 is the centralizer of an element of order 7 in M. • Finally, the Monster group itself is considered to be in this generation. (This series continues further: the product of M12 and a group of order 11 is the centralizer of an element of order 11 in M.) The Tits group, if regarded as a sporadic group, would belong in this generation: there is a subgroup S4 ×2F4(2)′ normalising a 2C2 subgroup of B, giving rise to a subgroup 2·S4 ×2F4(2)′ normalising a certain Q8 subgroup of the Monster. 2F4(2)′ is also a subquotient of the Fischer group Fi22, and thus also of Fi23 and Fi24′, and of the Baby Monster B. 2F4(2)′ is also a subquotient of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the ones already mentioned. Pariahs The six exceptions are J1, J3, J4, ''O'N, Ru, and Ly'', sometimes known as the pariahs. ==Table of the sporadic group orders (with Tits group)==