Sporadic group

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 who first predicted their existence. The full list is:

File:MonsterSporadicGroupGraph.svg|thumb|335px|The diagram shows the subquotient 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
Harada-Norton group HN or F₅₊ or F₅
Lyons group Ly
Thompson group Th or F_3|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."
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.
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 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
Co₂ is the stabilizer of a type 2 vector
Co₃ is the stabilizer of a type 3 vector
Suz is the group of automorphisms preserving a complex structure
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.
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
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
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.

The Tits group, if regarded as a sporadic group, would belong in this generation: there is a subgroup S₄ ×²F₄′ normalising a 2C² subgroup of B, giving rise to a subgroup 2·S₄ ×²F₄′ normalising a certain Q₈ subgroup of the Monster. ²F₄′ is also a subquotient of the Fischer group Fi₂₂, and thus also of Fi₂₃ and Fi₂₄′, and of the Baby Monster B. ²F₄′ is also a subquotient of the 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)

Group	Discoverer	Year	Order	Degree of minimal faithful Brauer character		Semi-presentation
M or F₁	Fischer, Griess	1973	808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 = 2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31·41·47·59·71 ≈ 8	196883	2A, 3B, 29
B or F₂	Fischer	1973	4,154,781,481,226,426,191,177,580,544,000,000 = 2⁴¹·3¹³·5⁶·7²·11·13·17·19·23·31·47 ≈ 4	4371	2C, 3A, 55
Fi₂₄ or F₃₊	Fischer	1971	1,255,205,709,190,661,721,292,800 = 2²¹·3¹⁶·5²·7³·11·13·17·23·29 ≈ 1	8671	2A, 3E, 29
Fi₂₃	Fischer	1971	4,089,470,473,293,004,800 = 2¹⁸·3¹³·5²·7·11·13·17·23 ≈ 4	782	2B, 3D, 28
Fi₂₂	Fischer	1971	64,561,751,654,400 = 2¹⁷·3⁹·5²·7·11·13 ≈ 6	78	2A, 13, 11
Th or F₃	Thompson	1976	90,745,943,887,872,000 = 2¹⁵·3¹⁰·5³·7²·13·19·31 ≈ 9	248	2, 3A, 19
Ly	Lyons	1972	51,765,179,004,000,000 = 2⁸·3⁷·5⁶·7·11·31·37·67 ≈ 5	2480	2, 5A, 14
HN or F₅	Harada, Norton	1976	273,030,912,000,000 = 2¹⁴·3⁶·5⁶·7·11·19 ≈ 3	133	2A, 3B, 22
Co₁	Conway	1969	4,157,776,806,543,360,000 = 2²¹·3⁹·5⁴·7²·11·13·23 ≈ 4	276	2B, 3C, 40
Co₂	Conway	1969	42,305,421,312,000 = 2¹⁸·3⁶·5³·7·11·23 ≈ 4	23	2A, 5A, 28
Co₃	Conway	1969	495,766,656,000 = 2¹⁰·3⁷·5³·7·11·23 ≈ 5	23	2A, 7C, 17
ON or O'N	O'Nan	1976	460,815,505,920 = 2⁹·3⁴·5·7³·11·19·31 ≈ 5	10944	2A, 4A, 11
Suz	Suzuki	1969	448,345,497,600 = 2¹³·3⁷·5²·7·11·13 ≈ 4	143	2B, 3B, 13
Ru	Rudvalis	1972	145,926,144,000 = 2¹⁴·3³·5³·7·13·29 ≈ 1	378	2B, 4A, 13
He or F₇	Held	1969	4,030,387,200 = 2¹⁰·3³·5²·7³·17 ≈ 4	51	2A, 7C, 17
McL	McLaughlin	1969	898,128,000 = 2⁷·3⁶·5³·7·11 ≈ 9	22	2A, 5A, 11
HS	Higman, Sims	1967	44,352,000 = 2⁹·3²·5³·7·11 ≈ 4	22	2A, 5A, 11
J₄	Janko	1976	86,775,571,046,077,562,880 = 2²¹·3³·5·7·11³·23·29·31·37·43 ≈ 9	1333	2A, 4A, 37
J₃ or HJM	Janko	1968	50,232,960 = 2⁷·3⁵·5·17·19 ≈ 5	85	2A, 3A, 19
J₂ or HJ	Janko	1968	604,800 = 2⁷·3³·5²·7 ≈ 6	14	2B, 3B, 7
J₁	Janko	1965	175,560 = 2³·3·5·7·11·19 ≈ 2	56	2, 3, 7
M₂₄	Mathieu	1861	244,823,040 = 2¹⁰·3³·5·7·11·23 ≈ 2	23	2B, 3A, 23
M₂₃	Mathieu	1861	10,200,960 = 2⁷·3²·5·7·11·23 ≈ 1	22	2, 4, 23
M₂₂	Mathieu	1861	443,520 = 2⁷·3²·5·7·11 ≈ 4	21	2A, 4A, 11
M₁₂	Mathieu	1861	95,040 = 2⁶·3³·5·11 ≈ 1	11	2B, 3B, 11
M₁₁	Mathieu	1861	7,920 = 2⁴·3²·5·11 ≈ 8	10	2, 4, 11
T or Ree group#Ree groups of type 2F4\|	Tits	1964	17,971,200 = 2¹¹·3³·5²·13 ≈ 2	104	2A, 3, 13