Short description: Finite simple set type not classified as Lie, cyclic or alternating

In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families[lower-alpha 1] plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group,[1] in which case there would be 27 sporadic groups.

The monster group, or friendly giant, is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it.[2]

## Names

Five of the sporadic groups were discovered by Émile Mathieu in the 1860s and the other twenty-one were found between 1965 and 1975. 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:[1][3][4]

The diagram shows the subquotient relations between the sporadic groups. A connecting line means the lower group is a subquotient of the upper, with no sporadic subquotient in between.
1st generation, File:EllipseSubqG.svg 2nd generation, File:EllipseSubqB.svg 3rd generation, File:EllipseSubqW.svg Pariah

Various constructions for these groups were first compiled in (Conway Curtis), 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 (Wilson Parker), updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic p ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in (Jansen 2005).

A further exception in the classification of finite simple groups is the Tits group T, which is sometimes considered of Lie type[5] or sporadic — it is almost but not strictly a group of Lie type[6] — which is why in some sources the number of sporadic groups is given as 27, instead of 26.[7][8] In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both.[9][citation needed] The Tits group is the (n = 0)-member 2F4(2)′ of the infinite family of commutator groups 2F4(22n+1)′; thus in a strict sense not sporadic, nor of Lie type. For n > 0 these finite simple groups coincide with the groups of Lie type 2F4(22n+1), also known as Ree groups of type 2F4.

The earliest use of the term sporadic group may be (Burnside 1911) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received."

The diagram at right is based on (Ronan 2006). 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.[10][lower-alpha 2]

#### 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.[11]

#### Second generation (7 groups): the Leech lattice

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:[12]

• 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 6) 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:[13]

• 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.[14][15]

## Table of the sporadic group orders (with Tits group)

Group Discoverer [16]
Year
Generation
[4][17]
Order
[1][4]
Factorized order
[18]
Minimal faithful Brauer character
[19][20]
$\displaystyle{ (a, b, ab) }$
Generators
[20][lower-alpha 3]
$\displaystyle{ \langle\langle a,b \mid o(z)\rangle\rangle }$
Semi-presentation
M or F1 Fischer, Griess 1973 3 3rd align=right | 808017424794512875886459904961710757005754368000000000 ≈ 8×1053 246 × 320 × 59 × 76 × 112 ×  133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71 196883 2A, 3B, 29 $\displaystyle{ o\bigl((ab)^{4}(ab^{2})^{2}\bigr) = 50 }$
B or F2 Fischer 1973 3 3rd align=right | 4154781481226426191177580544000000 ≈ 4×1033 241 × 313 × 56 × 72 × 11 × 13 × 17 × 19 × 23 × 31 × 47 4371 2C, 3A, 55 $\displaystyle{ o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 23 }$
Fi24 or F3+ Fischer 1971 3 3rd align=right | 1255205709190661721292800 ≈ 1×1024 221 × 316 × 52 × 73 × 11 × 13 × 17 × 23 × 29 8671 2A, 3E, 29 $\displaystyle{ o\bigl((ab)^3 b\bigr) = 33 }$
Fi23 Fischer 1971 3 3rd align=right | 4089470473293004800 ≈ 4×1018 218 × 313 × 52 × 7 × 11 × 13 × 17 × 23 782 2B, 3D, 28 $\displaystyle{ o\bigl(a^{bb}(ab)^{14}\bigr) = 5 }$
Fi22 Fischer 1971 3 3rd align=right | 64561751654400 ≈ 6×1013 217 × 39 × 52 × 7 × 11 × 13 78 2A, 13, 11 $\displaystyle{ o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 12 }$
Th or F3 Thompson 1976 3 3rd align=right | 90745943887872000 ≈ 9×1016 215 × 310 × 53 × 72 × 13 × 19 × 31 248 2, 3A, 19 $\displaystyle{ o\bigl((ab)^{3}b\bigr) = 21 }$
Ly Lyons 1972 4 Pariah align=right | 51765179004000000 ≈ 5×1016 28 × 37 × 56 × 7 × 11 × 31 × 37 × 67 2480 2, 5A, 14 $\displaystyle{ o\bigl(ababab^2\bigr) = 67 }$
HN or F5 Harada, Norton 1976 3 3rd align=right | 273030912000000 ≈ 3×1014 214 × 36 × 56 × 7 × 11 × 19 133 2A, 3B, 22 $\displaystyle{ o\bigl([a, b]\bigr) = 5 }$
Co1 Conway 1969 2 2nd align=right | 4157776806543360000 ≈ 4×1018 221 × 39 × 54 × 72 × 11 × 13 × 23 276 2B, 3C, 40 $\displaystyle{ o\bigl(ab(abab^{2})^{2}\bigr) = 42 }$
Co2 Conway 1969 2 2nd align=right | 42305421312000 ≈ 4×1013 218 × 36 × 53 × 7 × 11 × 23 23 2A, 5A, 28 $\displaystyle{ o\bigl([a,b]\bigr) = 4 }$
Co3 Conway 1969 2 2nd align=right | 495766656000 ≈ 5×1011 210 × 37 × 53 × 7 × 11 × 23 23 2A, 7C, 17 $\displaystyle{ o\bigl((uvv)^{3}(uv)^{6}\bigr) = 5 }$[lower-alpha 4]
ON or O'N O'Nan 1976 4 Pariah align=right | 460815505920 ≈ 5×1011 29 × 34 × 5 × 73 × 11 × 19 × 31 10944 2A, 4A, 11 $\displaystyle{ o\bigl(abab(b^{2}(b^{2})^{abab})^{5}\bigr) = 5 }$
Suz Suzuki 1969 2 2nd align=right | 448345497600 ≈ 4×1011 213 × 37 × 52 × 7 × 11 × 13 143 2B, 3B, 13 $\displaystyle{ o\bigl([a, b]\bigr) = 15 }$
Ru Rudvalis 1972 4 Pariah align=right | 145926144000 ≈ 1×1011 214 × 33 × 53 × 7 × 13 × 29 378 2B, 4A, 13 $\displaystyle{ o(abab^{2}) = 29 }$
He or F7 Held 1969 3 3rd align=right | 4030387200 ≈ 4×109 210 × 33 × 52 × 73 × 17 51 2A, 7C, 17 $\displaystyle{ o\bigl(ab^{2}abab^{2}ab^{2}\bigr) = 10 }$
McL McLaughlin 1969 2 2nd align=right | 898128000 ≈ 9×108 27 × 36 × 53 × 7 × 11 22 2A, 5A, 11 $\displaystyle{ o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 7 }$
HS Higman, Sims 1967 2 2nd align=right | 44352000 ≈ 4×107 29 × 32 × 53 × 7 × 11 22 2A, 5A, 11 $\displaystyle{ o(abab^{2}) = 15 }$
J4 Janko 1976 4 Pariah align=right | 86775571046077562880 ≈ 9×1019 221 × 33 × 5 × 7 × 113 × 23 × 29 × 31 × 37 × 43 1333 2A, 4A, 37 $\displaystyle{ o\bigl(abab^2\bigr) = 10 }$
J3 or HJM Janko 1968 4 Pariah align=right | 50232960 ≈ 5×107 27 × 35 × 5 × 17 × 19 85 2A, 3A, 19 $\displaystyle{ o\bigl([a, b]\bigr) = 9 }$
J2 or HJ Janko 1968 2 2nd align=right | 604800 ≈ 6×105 27 × 33 × 52 × 7 14 2B, 3B, 7 $\displaystyle{ o\bigl([a, b]\bigr) = 12 }$
J1 Janko 1965 4 Pariah align=right | 175560 ≈ 2×105 23 × 3 × 5 × 7 × 11 × 19 56 2, 3, 7 $\displaystyle{ o\bigl(abab^2\bigr) = 19 }$
T or 2F4(2)′ Tits 1964 3 3rd align=right | 17971200 ≈ 2×107 211 × 33 × 52 × 13 104[21] 2A, 3, 13 $\displaystyle{ o\bigl([a, b]\bigr) = 5 }$
M24 Mathieu 1861 1 1st align=right | 244823040 ≈ 2×108 210 × 33 × 5 × 7 × 11 × 23 23 2B, 3A, 23 $\displaystyle{ o\bigl(ab(abab^2)^2 ab^2\bigr) = 4 }$
M23 Mathieu 1861 1 1st align=right | 10200960 ≈ 1×107 27 × 32 × 5 × 7 × 11 × 23 22 2, 4, 23 $\displaystyle{ o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 8 }$
M22 Mathieu 1861 1 1st align=right |443520 ≈ 4×105 27 × 32 × 5 × 7 × 11 21 2A, 4A, 11 $\displaystyle{ o\bigl(abab^2\bigr) = 11 }$
M12 Mathieu 1861 1 1st align=right |95040 ≈ 1×105 26 × 33 × 5 × 11 11 2B, 3B, 11 $\displaystyle{ o\bigl([a,b]\bigr) = o\bigl(ababab^{2}\bigr) = 6 }$
M11 Mathieu 1861 1 1st align=right |7920 ≈ 8×103 24 × 32 × 5 × 11 10 2, 4, 11 $\displaystyle{ o\bigl((ab)^2 (abab^2)^2 ab^2\bigr) = 4 }$

## Notes

1. The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups 2F4(22n+1)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
2. (Conway Curtis) organizes the 26 sporadic groups in likeness:
"The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the half-dozen oddments."
3. Here listed are semi-presentations from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed.
4. Where $\displaystyle{ u = (b^{2}(b^{2})abb)^{3} }$ and $\displaystyle{ v = t(b^{2}(b^{2})t)^{2} }$ with $\displaystyle{ t = abab^{3}a^{2} }$.

## References

1. (Conway Curtis)
2. (Griess, Jr. 1998)
3. (Gorenstein Lyons)
4. (Ronan 2006)
5. (Howlett Rylands)
"This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group)."
6. (Gorenstein 1979)
7. (Conway Curtis)
8. (Hartley Hulpke)
"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group 2F4(2)′ is counted also) which these infinite families do not include."
9. (Wilson Parker)
10. (Griess, Jr. 1982)
11. (Griess, Jr. 1998)
12. (Griess, Jr. 1998)
13. (Griess, Jr. 1998)
14. (Griess, Jr. 1982)
15. (Griess, Jr. 1998)
16. (Hiss 2003)
Tabelle 2. Die Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)
17. (Sloane 1996)
18. (Jansen 2005)
19. (Nickerson Wilson)
20. (Wilson Parker)
21. (Lubeck 2001)

### Works cited

• Howlett, R. B.; Rylands, L. J.; Taylor, D. E. (2001). "Matrix generators for exceptional groups of Lie type". Journal of Symbolic Computation 31 (4): 429–445. doi:10.1006/jsco.2000.0431.