# Sporadic group

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In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.
A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The mentioned classification theorem states that the list of finite simple groups consists of 18 countably infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. The Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, 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.

## 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:
* Mathieu groups M$_{11}$, M$_{12}$, M$_{22}$, M$_{23}$, M$_{24}$
* Janko groups J$_1$, J$_2$ or HJ, J$_3$ or HJM, J$_4$
* Conway groups Co$_1$, Co$_2$, Co$_3$
* Fischer groups Fi$_{22}$, Fi$_{23}$, Fi$_{24}$′ or F$_{3+}$
* Higman-Sims group HS
* McLaughlin group McL
* Held group He or F$_{7+}$ or F$_7$
* Rudvalis group Ru
* Suzuki group Suz or F$_{3−}$
* O'Nan group O'N (ON)
* Harada-Norton group HN or F$_{5+}$ or F$_5$
* Lyons group Ly
* Thompson group Th or F$_{3|3}$ or F$_3$
* Baby Monster group B or F$_{2+}$ or F$_2$
* Fischer-Griess Monster group M or F$_1$
Various constructions for these groups were first compiled in Conway et al. (1985), 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 et al. (1999), 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 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 (n = 0)-member $^2$F$_4$(2)′ of the infinite family of commutator groups $^2$F$_4$(2$^{2n+1}$)′; 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 $^2$F$_4$(2$^{2n+1}$), also known as Ree groups of type $^2$F$_4$.
The earliest use of the term sporadic group may be Burnside (1911, p. 504) 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 at right is based on Ronan (2006, p. 247). 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

M$_n$ for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M$_{24}$, 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:
* Co$_1$ is the quotient of the automorphism group by its center {±1}
* Co$_2$ is the stabilizer of a type 2 (i.e., length 2) vector
* Co$_3$ 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
* J$_2$ 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 F$_2$ has a double cover which is the centralizer of an element of order 2 in M
* Fi$_{24}$′ has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A")
* Fi$_{23}$ is a subgroup of Fi$_{24}$′
* Fi$_{22}$ has a double cover which is a subgroup of Fi$_{23}$
* The product of Th = F$_3$ 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 = F$_5$ and a group of order 5 is the centralizer of an element of order 5 in M
* The product of He = F$_7$ 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 M$_{12}$ 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 S$_4$ ×$^2$F$_4$(2)′ normalising a 2C$^2$ subgroup of B, giving rise to a subgroup 2·S$_4$ ×$^2$F$_4$(2)′ normalising a certain Q$_8$ subgroup of the Monster. $^2$F$_4$(2)′ is also a subquotient of the Fischer group Fi$_{22}$, and thus also of Fi$_{23}$ and Fi$_{24}$′, and of the Baby Monster B. $^2$F$_4$(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 J$_1$, J$_3$, J$_4$, O'N, Ru, and Ly, sometimes known as the pariahs.

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

| Group | Discoverer | Year | Generation | Order | Degree of minimal faithful Brauer character | $(a,b,ab)$ Generators | $\langle \langle a,b\mid o(z)\rangle \rangle$ Semi-presentation |
| M or F$_1$ | Fischer, Griess | 1973 | 3rd | 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 = 2$^{46}$·3$^{20}$·5$^9$·7$^6$·11$^2$·13$^3$·17·19·23·29·31·41·47·59·71 ≈ 8×10$^{53}$ | 196883 | 2A, 3B, 29 | $o{\bigl(ab)^{4}(ab^{2})^{2}{\bigr )}=50$ |
| B or F$_2$ | Fischer | 1973 | 3rd | 4,154,781,481,226,426,191,177,580,544,000,000 = 2$^{41}$·3$^{13}$·5$^6$·7$^2$·11·13·17·19·23·31·47 ≈ 4×10$^{33}$ | 4371 | 2C, 3A, 55 | $o{\bigl(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=23$ |
| Fi$_{24}$ or F$_{3+}$ | Fischer | 1971 | 3rd | 1,255,205,709,190,661,721,292,800 = 2$^{21}$·3$^{16}$·5$^2$·7$^3$·11·13·17·23·29 ≈ 1×10$^{24}$ | 8671 | 2A, 3E, 29 | $o{\bigl(ab)^{3}b{\bigr )}=33$ |
| Fi$_{23}$ | Fischer | 1971 | 3rd | 4,089,470,473,293,004,800 = 2$^{18}$·3$^{13}$·5$^2$·7·11·13·17·23 ≈ 4×10$^{18}$ | 782 | 2B, 3D, 28 | $o{\bigla^{bb}(ab)^{14}{\bigr )}=5$ |
| Fi$_{22}$ | Fischer | 1971 | 3rd | 64,561,751,654,400 = 2$^{17}$·3$^9$·5$^2$·7·11·13 ≈ 6×10$^{13}$ | 78 | 2A, 13, 11 | $o{\bigl(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=12$ |
| Th or F$_3$ | Thompson | 1976 | 3rd | 90,745,943,887,872,000 = 2$^{15}$·3$^{10}$·5$^3$·7$^2$·13·19·31 ≈ 9×10$^{16}$ | 248 | 2, 3A, 19 | $o{\bigl(ab)^{3}b{\bigr )}=21$ |
| Ly | Lyons | 1972 | Pariah | 51,765,179,004,000,000 = 2$^8$·3$^7$·5$^6$·7·11·31·37·67 ≈ 5×10$^{16}$ | 2480 | 2, 5A, 14 | $o{\biglababab^{2}{\bigr )}=67$ |
| HN or F$_5$ | Harada, Norton | 1976 | 3rd | 273,030,912,000,000 = 2$^{14}$·3$^6$·5$^6$·7·11·19 ≈ 3×10$^{14}$ | 133 | 2A, 3B, 22 | $o{\bigl[a,b]{\bigr )}=5$ |
| Co$_1$ | Conway | 1969 | 2nd | 4,157,776,806,543,360,000 = 2$^{21}$·3$^9$·5$^4$·7$^2$·11·13·23 ≈ 4×10$^{18}$ | 276 | 2B, 3C, 40 | $o{\biglab(abab^{2})^{2}{\bigr )}=42$ |
| Co$_2$ | Conway | 1969 | 2nd | 42,305,421,312,000 = 2$^{18}$·3$^6$·5$^3$·7·11·23 ≈ 4×10$^{13}$ | 23 | 2A, 5A, 28 | $o{\bigl[a,b]{\bigr )}=4$ |
| Co$_3$ | Conway | 1969 | 2nd | 495,766,656,000 = 2$^{10}$·3$^7$·5$^3$·7·11·23 ≈ 5×10$^{11}$ | 23 | 2A, 7C, 17 | $o{\bigl(uvv)^{3}(uv)^{6}{\bigr )}=5$ |
| ON or O'N | O'Nan | 1976 | Pariah | 460,815,505,920 = 2$^9$·3$^4$·5·7$^3$·11·19·31 ≈ 5×10$^{11}$ | 10944 | 2A, 4A, 11 | $o{\biglabab(b^{2}(b^{2})^{abab})^{5}{\bigr )}=5$ |
| Suz | Suzuki | 1969 | 2nd | 448,345,497,600 = 2$^{13}$·3$^7$·5$^2$·7·11·13 ≈ 4×10$^{11}$ | 143 | 2B, 3B, 13 | $o{\bigl[a,b]{\bigr )}=15$ |
| Ru | Rudvalis | 1972 | Pariah | 145,926,144,000 = 2$^{14}$·3$^3$·5$^3$·7·13·29 ≈ 1×10$^{11}$ | 378 | 2B, 4A, 13 | $o(abab^{2})=29$ |
| He or F$_7$ | Held | 1969 | 3rd | 4,030,387,200 = 2$^{10}$·3$^3$·5$^2$·7$^3$·17 ≈ 4×10$^9$ | 51 | 2A, 7C, 17 | $o{\biglab^{2}abab^{2}ab^{2}{\bigr )}=10$ |
| McL | McLaughlin | 1969 | 2nd | 898,128,000 = 2$^7$·3$^6$·5$^3$·7·11 ≈ 9×10$^8$ | 22 | 2A, 5A, 11 | $o{\bigl(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=7$ |
| HS | Higman, Sims | 1967 | 2nd | 44,352,000 = 2$^9$·3$^2$·5$^3$·7·11 ≈ 4×10$^7$ | 22 | 2A, 5A, 11 | $o(abab^{2})=15$ |
| J$_4$ | Janko | 1976 | Pariah | 86,775,571,046,077,562,880 = 2$^{21}$·3$^3$·5·7·11$^3$·23·29·31·37·43 ≈ 9×10$^{19}$ | 1333 | 2A, 4A, 37 | $o{\biglabab^{2}{\bigr )}=10$ |
| J$_3$ or HJM | Janko | 1968 | Pariah | 50,232,960 = 2$^7$·3$^5$·5·17·19 ≈ 5×10$^7$ | 85 | 2A, 3A, 19 | $o{\bigl[a,b]{\bigr )}=9$ |
| J$_2$ or HJ | Janko | 1968 | 2nd | 604,800 = 2$^7$·3$^3$·5$^2$·7 ≈ 6×10$^5$ | 14 | 2B, 3B, 7 | $o{\bigl[a,b]{\bigr )}=12$ |
| J$_1$ | Janko | 1965 | Pariah | 175,560 = 2$^3$·3·5·7·11·19 ≈ 2×10$^5$ | 56 | 2, 3, 7 | $o{\biglabab^{2}{\bigr )}=19$ |
| M$_{24}$ | Mathieu | 1861 | 1st | 244,823,040 = 2$^{10}$·3$^3$·5·7·11·23 ≈ 2×10$^8$ | 23 | 2B, 3A, 23 | $o{\biglab(abab^{2})^{2}ab^{2}{\bigr )}=4$ |
| M$_{23}$ | Mathieu | 1861 | 1st | 10,200,960 = 2$^7$·3$^2$·5·7·11·23 ≈ 1×10$^7$ | 22 | 2, 4, 23 | $o{\bigl(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=8$ |
| M$_{22}$ | Mathieu | 1861 | 1st | 443,520 = 2$^7$·3$^2$·5·7·11 ≈ 4×10$^5$ | 21 | 2A, 4A, 11 | $o{\biglabab^{2}{\bigr )}=11$ |
| M$_{12}$ | Mathieu | 1861 | 1st | 95,040 = 2$^6$·3$^3$·5·11 ≈ 1×10$^5$ | 11 | 2B, 3B, 11 | $o{\bigl[a,b]{\bigr )}=o{\biglababab^{2}{\bigr )}=6$ |
| M$_{11}$ | Mathieu | 1861 | 1st | 7,920 = 2$^4$·3$^2$·5·11 ≈ 8×10$^3$ | 10 | 2, 4, 11 | $o{\bigl(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=4$ |
| T or $^2$F$_4$(2)′ | Tits | 1964 | 3rd | 17,971,200 = 2$^{11}$·3$^3$·5$^2$·13 ≈ 2×10$^7$ | 104 | 2A, 3, 13 | $o{\bigl[a,b]{\bigr )}=5$ |