List of finite simple groups
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates.
Summary
The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. and A2 both have order 20160, and that the group Bn has the same order as Cn for q odd, n > 2. The smallest of the latter pairs of groups are B3 and C3There is an unfortunate conflict between the notations for the alternating groups An and the groups of Lie type An. Some authors use various different fonts for An to distinguish them. In particular,
in this article we make the distinction by setting the alternating groups An in Roman font and the Lie-type groups An in italic.
In what follows, n is a positive integer, and q is a positive power of a prime number p, with the restrictions noted. The notation represents the greatest common divisor of the integers a and b.
[Cyclic group]s, Z''p''
Simplicity: Simple for p a prime number.Order: p
Schur multiplier: Trivial.
Outer automorphism group: Cyclic of order p − 1.
Other names: Z/pZ, Cp
Remarks: These are the only simple groups that are not perfect.
[Alternating group]s, A''n'', ''n'' > 4
Simplicity: Solvable for n < 5, otherwise simple.Order: n!/2 when n > 1.
Schur multiplier: 2 for n = 5 or n > 7, 6 for n = 6 or 7; see Covering groups of the alternating and symmetric groups
Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for n = 6, it has order 4.
Other names: Altn.
Isomorphisms: A1 and A2 are trivial. A3 is cyclic of order 3. A4 is isomorphic to A1. A5 is isomorphic to A1 and to A1. A6 is isomorphic to A1 and to the derived group B2′. A8 is isomorphic to A3.
Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1.
Groups of Lie type
Notation: n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d⋅f⋅g, where d is the order of the group of "diagonal automorphisms", f is the order of the group of "field automorphisms", and g is the order of the group of "graph automorphisms". The outer automorphism group is isomorphic to the semidirect product where all these groups are cyclic of the respective orders d, f, g, except for type, odd, where the group of order is, and , the symmetric group on three elements. The notation represents the greatest common divisor of the integers a and b.[Chevalley group]s, ''An''(''q''), ''Bn''(''q'') ''n'' > 1, ''Cn''(''q'') ''n'' > 2, ''Dn''(''q'') ''n'' > 3
| Chevalley groups, An linear groups | Chevalley groups, Bn n > 1 orthogonal groups | Chevalley groups, Cn n > 2 symplectic groups | Chevalley groups, Dn n > 3 orthogonal groups | |
| Simplicity | A1 and A1 are solvable, the others are simple. | B2 is not simple but its derived group B2′ is a simple subgroup of index 2; the others are simple. | All simple | All simple |
| Order | ||||
| Schur multiplier | For the simple groups it is cyclic of order except for A1, A1, A2, A2, A3. | except for B2 = S6, order 6 for B2 and B3 and B3. | except for C3. | The order is except for D4. |
| Outer automorphism group | ⋅f⋅1 for n = 1; ⋅f⋅2 for n > 1, where q = pf | ⋅f⋅1 for q odd or n > 2; ⋅f⋅2 for q even and n = 2, where q = pf | ⋅f⋅1, where q = pf | 2⋅f⋅S3 for n = 4, 2⋅f⋅2 for n > 4 even, ⋅f⋅2 for n odd, where q = pf, and S3 is the symmetric group of order 3! on 3 points. |
| Other names | Projective special linear groups, PSLn+1, Ln+1, PSL | O2n+1, Ω2n+1. | Projective symplectic group, PSp2n, PSpn, S2n, Abelian group. | O2n+, PΩ2n+. "Hypoabelian group" is an archaic name for this group in characteristic 2. |
| Isomorphisms | A1 is isomorphic to the symmetric group on 3 points of order 6. A1 is isomorphic to the alternating group A4. A1 and A1 are both isomorphic to the alternating group A5. A1 and A2 are isomorphic. A1 is isomorphic to the derived group 2G2′. A1 is isomorphic to A6 and to the derived group B2′. A3 is isomorphic to A8. | Bn is isomorphic to Cn. B2 is isomorphic to the symmetric group on 6 points, and the derived group B2′ is isomorphic to A1 and to A6. B2 is isomorphic to 2A3. | Cn is isomorphic to Bn | |
| Remarks | These groups are obtained from the general linear groups GLn+1 by taking the elements of determinant 1 and then quotienting out by the center. | This is the group obtained from the orthogonal group in dimension 2n + 1 by taking the kernel of the determinant and spinor norm maps. B1 also exists, but is the same as A1. B2 has a non-trivial graph automorphism when q is a power of 2. | This group is obtained from the symplectic group in 2n dimensions by quotienting out the center. C1 also exists, but is the same as A1. C2 also exists, but is the same as B2. | This is the group obtained from the split orthogonal group in dimension 2n by taking the kernel of the determinant and spinor norm maps and then killing the center. The groups of type D4 have an unusually large diagram automorphism group of order 6, containing the triality automorphism. D2 also exists, but is the same as A1×A1. D3 also exists, but is the same as A3. |
[Chevalley group]s, ''E''6(''q''), ''E''7(''q''), ''E''8(''q''), ''F''4(''q''), ''G''2(''q'')
| Chevalley groups, E6 | Chevalley groups, E7 | Chevalley groups, E8 | Chevalley groups, F4 | Chevalley groups, G2 | |
| Simplicity | All simple | All simple | All simple | All simple | G2 is not simple but its derived group G2′ is a simple subgroup of index 2; the others are simple. |
| Order | q36/ | q63/ | q120 | q24 | q6 |
| Schur multiplier | Trivial | Trivial except for F4 | Trivial for the simple groups except for G2 and G2 | ||
| Outer automorphism group | ⋅f⋅2, where q = pf | ⋅f⋅1, where q = pf | 1⋅f⋅1, where q = pf | 1⋅f⋅1 for q odd, 1⋅f⋅2 for q even, where q = pf | 1⋅f⋅1 for q not a power of 3, 1⋅f⋅2 for q a power of 3, where q = pf |
| Other names | Exceptional Chevalley group | Exceptional Chevalley group | Exceptional Chevalley group | Exceptional Chevalley group | Exceptional Chevalley group |
| Isomorphisms | The derived group G2′ is isomorphic to 2A2. | ||||
| Remarks | Has two representations of dimension 27, and acts on the Lie algebra of dimension 78. | Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133. | It acts on the corresponding Lie algebra of dimension 248. E8 contains the Thompson simple group. | These groups act on 27-dimensional exceptional Jordan algebras, which gives them 26-dimensional representations. They also act on the corresponding Lie algebras of dimension 52. F4 has a non-trivial graph automorphism when q is a power of 2. | These groups are the automorphism groups of 8-dimensional Cayley algebras over finite fields, which gives them 7-dimensional representations. They also act on the corresponding Lie algebras of dimension 14. G2 has a non-trivial graph automorphism when q is a power of 3. Moreover, they appear as automorphism groups of certain point-line geometries called split Cayley generalized hexagons. |
Steinberg group">Group of Lie type#Steinberg groups">Steinberg groups, 2''An''(''q''2) ''n'' > 1, 2''Dn''(''q''2) ''n'' > 3, 2''E''6(''q''2), 3''D''4(''q''3)
| Steinberg groups, 2An n > 1 unitary groups | Steinberg groups, 2Dn n > 3 orthogonal groups | Steinberg groups, 2E6 | Steinberg groups, 3D4 | |
| Simplicity | 2A2 is solvable, the others are simple. | All simple | All simple | All simple |
| Order | q36/ | q12 | ||
| Schur multiplier | Cyclic of order for the simple groups, except for 2A3, 2A3, 2A5 | Cyclic of order | except for 2E6. | Trivial |
| Outer automorphism group | ⋅f⋅1, where q2 = pf | ⋅f⋅1, where q2 = pf | ⋅f⋅1, where q2 = pf | 1⋅f⋅1, where q3 = pf |
| Other names | Twisted Chevalley group, projective special unitary group, PSUn+1, PSU, Un+1, 2An, 2An | 2Dn, O2n−, PΩ2n−, twisted Chevalley group. "Hypoabelian group" is an archaic name for this group in characteristic 2. | 2E6, twisted Chevalley group | 3D4, D42, Twisted Chevalley groups |
| Isomorphisms | The solvable group 2A2 is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. 2A2 is isomorphic to the derived group G2′. 2A3 is isomorphic to B2. | |||
| Remarks | This is obtained from the unitary group in n + 1 dimensions by taking the subgroup of elements of determinant 1 and then quotienting out by the center. | This is the group obtained from the non-split orthogonal group in dimension 2n by taking the kernel of the determinant and spinor norm maps and then killing the center. 2D2 also exists, but is the same as A1. 2D3 also exists, but is the same as 2A3. | One of the exceptional double covers of 2E6 is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group. | 3D4 acts on the unique even 26-dimensional lattice of determinant 3 with no roots. |
[Suzuki groups], 2''B''2(22''n''+1)
Simplicity: Simple for n ≥ 1. The group2B2 is solvable.
Order:
q2
,
where
q = 22n+1.
Schur multiplier: Trivial for n ≠ 1, elementary abelian of order 4
for 2B2.
Outer automorphism group:
where f = 2n + 1.
Other names: Suz, Sz.
Isomorphisms: 2B2 is the Frobenius group of order 20.
Remarks: Suzuki group are Zassenhaus groups acting on sets of size 2 + 1, and have 4-dimensional representations over the field with 22n+1 elements. They are the only non-cyclic simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group.
[Ree group]s and [Tits group], 2''F''4(22''n''+1)
Simplicity: Simple for n ≥ 1. The derived group 2F4′ is simple of index 2in 2F4, and is called the Tits group,
named for the Belgian mathematician Jacques Tits.
Order:
q12
,
where
q = 22n+1.
The Tits group has order 17971200 = 211 ⋅ 33 ⋅ 52 ⋅ 13.
Schur multiplier: Trivial for n ≥ 1 and for the Tits group.
Outer automorphism group:
where f = 2n + 1. Order 2 for the Tits group.
Remarks: Unlike the other simple groups of Lie type, the Tits group does not have a BN pair, though its automorphism group does so most authors count it as a sort of honorary group of Lie type.
[Ree group]s, 2''G''2(32''n''+1)
Simplicity: Simple for n ≥ 1. The group 2G2 is not simple, but its derived group 2G2′ is a simple subgroup of index 3.Order:
q3
,
where
q = 32n+1
Schur multiplier: Trivial for n ≥ 1 and for 2G2′.
Outer automorphism group:
where f = 2n + 1.
Other names: Ree, R, E2∗.
Isomorphisms: The derived group 2G2′ is isomorphic to A1.
Remarks: 2G2 has a doubly transitive permutation representation on 33 + 1 points and acts on a 7-dimensional vector space over the field with 32n+1 elements.
Sporadic groups
[Mathieu group]s, M11, M12, M22, M23, M24
[Janko group]s, J1, J2, J3, J4
| Janko group, J1 | Janko group, J2 | Janko group, J3 | Janko group, J4 | |
| Order | 23 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19 = 175560 | 27 ⋅ 33 ⋅ 52 ⋅ 7 = 604800 | 27 ⋅ 35 ⋅ 5 ⋅ 17 ⋅ 19 = 50232960 | 221 ⋅ 33 ⋅ 5 ⋅ 7 ⋅ 113 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 37 ⋅ 43 = 86775571046077562880 |
| Schur multiplier | Trivial | Order 2 | Order 3 | Trivial |
| Outer automorphism group | Trivial | Order 2 | Order 2 | Trivial |
| Other names | J, J | Hall–Janko group, HJ | Higman–Janko–McKay group, HJM | |
| Remarks | It is a subgroup of G2, and so has a 7-dimensional representation over the field with 11 elements. | The automorphism group J2:2 of J2 is the automorphism group of a rank 3 graph on 100 points called the Hall-Janko graph. It is also the automorphism group of a regular near octagon called the Hall-Janko near octagon. The group J2 is contained in G2. | J3 seems unrelated to any other sporadic groups. Its triple cover has a 9-dimensional unitary representation over the field with 4 elements. | Has a 112-dimensional representation over the field with 2 elements. |
[Conway group]s, Co1, Co2, Co3
| Conway group, Co1 | Conway group, Co2 | Conway group, Co3 | |
| Order | 221 ⋅ 39 ⋅ 54 ⋅ 72 ⋅ 11 ⋅ 13 ⋅ 23 = 4157776806543360000 | 218 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 ⋅ 23 = 42305421312000 | 210 ⋅ 37 ⋅ 53 ⋅ 7 ⋅ 11 ⋅ 23 = 495766656000 |
| Schur multiplier | Order 2 | Trivial | Trivial |
| Outer automorphism group | Trivial | Trivial | Trivial |
| Other names | ·1 | ·2 | ·3, C3 |
| Remarks | The perfect double cover Co0 of Co1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0. | Subgroup of Co0; fixes a norm 4 vector in the Leech lattice. | Subgroup of Co0; fixes a norm 6 vector in the Leech lattice. It has a doubly transitive permutation representation on 276 points. |
[Fischer group]s, Fi22, Fi23, Fi24′
| Fischer group, Fi22 | Fischer group, Fi23 | Fischer group, Fi24′ | |
| Order | 217 ⋅ 39 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 = 64561751654400 | 218 ⋅ 313 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 23 = 4089470473293004800 | 221 ⋅ 316 ⋅ 52 ⋅ 73 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 23 ⋅ 29 = 1255205709190661721292800 |
| Schur multiplier | Order 6 | Trivial | Order 3 |
| Outer automorphism group | Order 2 | Trivial | Order 2 |
| Other names | M | M | M′, F3+ |
| Remarks | A 3-transposition group whose double cover is contained in Fi23. | A 3-transposition group contained in Fi24′. | The triple cover is contained in the monster group. |
[Higman–Sims group], HS
Order: 29 ⋅ 32 ⋅ 53 ⋅ 7 ⋅ 11 = 44352000Schur multiplier: Order 2.
Outer automorphism group: Order 2.
Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co2 and in Co3.
McLaughlin group">McLaughlin group (mathematics)">McLaughlin group, McL
Order: 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898128000Schur multiplier: Order 3.
Outer automorphism group: Order 2.
Remarks: Acts as a rank 3 permutation group on the McLaughlin graph with 275 points, and is contained in Co2 and in Co3.
[Held group], He
Order:210 ⋅ 33 ⋅ 52 ⋅ 73 ⋅ 17 = 4030387200
Schur multiplier: Trivial.
Outer automorphism group: Order 2.
Other names: Held–Higman–McKay group, HHM, F7, HTH
Remarks: Centralizes an element of order 7 in the monster group.
[Rudvalis group], Ru
Order:214 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 13 ⋅ 29 = 145926144000
Schur multiplier: Order 2.
Outer automorphism group: Trivial.
Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers.
[Suzuki sporadic group], Suz
Order: 213 ⋅ 37 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 = 448345497600Schur multiplier: Order 6.
Outer automorphism group: Order 2.
Other names: Sz
Remarks: The 6 fold cover acts on a 12-dimensional lattice over the Eisenstein integers. It is not related to the Suzuki groups of Lie type.
[O'Nan group], O'N
Order:29 ⋅ 34 ⋅ 5 ⋅ 73 ⋅ 11 ⋅ 19 ⋅ 31 = 460815505920
Schur multiplier: Order 3.
Outer automorphism group: Order 2.
Other names: O'Nan–Sims group, O'NS, O–S
Remarks:
The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.
[Harada–Norton group], HN
Order:214 ⋅ 36 ⋅ 56 ⋅ 7 ⋅ 11 ⋅ 19 = 273030912000000
Schur multiplier: Trivial.
Outer automorphism group: Order 2.
Other names: F5, D
Remarks: Centralizes an element of order 5 in the monster group.
[Lyons group], Ly
Order:28 ⋅ 37 ⋅ 56 ⋅ 7 ⋅ 11 ⋅ 31 ⋅ 37 ⋅ 67 = 51765179004000000
Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Other names: Lyons–Sims group, LyS
Remarks: Has a 111-dimensional representation over the field with 5 elements.
Thompson group">Thompson group (finite)">Thompson group, Th
Order: 215 ⋅ 310 ⋅ 53 ⋅ 72 ⋅ 13 ⋅ 19 ⋅ 31 = 90745943887872000Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Other names: F3, E
Remarks: Centralizes an element of order 3 in the monster, and is contained in E8, so has a 248-dimensional representation over the field with 3 elements.
[Baby Monster group], B
Order:Schur multiplier: Order 2.
Outer automorphism group: Trivial.
Other names: F2
Remarks: The double cover is contained in the monster group. It has a representation of dimension 4371 over the complex numbers, and a representation of dimension 4370 over the field with 2 elements preserving a commutative but non-associative product.
Fischer–Griess [Monster group], M
Order:Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Other names: F1, M1, Monster group, Friendly giant, Fischer's monster.
Remarks: Contains all but 6 of the other sporadic groups as subquotients. Related to monstrous moonshine. The monster is the automorphism group of the 196,883-dimensional Griess algebra and the infinite-dimensional monster vertex operator algebra, and acts naturally on the monster Lie algebra.
Non-cyclic simple groups of small order
| Order | Factored order | Group | Schur multiplier | Outer automorphism group |
| 60 | 22 ⋅ 3 ⋅ 5 | A5 = A1 = A1 | 2 | 2 |
| 168 | 23 ⋅ 3 ⋅ 7 | A1 = A2 | 2 | 2 |
| 360 | 23 ⋅ 32 ⋅ 5 | A6 = A1 = B2′ | 6 | 2×2 |
| 504 | 23 ⋅ 32 ⋅ 7 | A1 = 2G2′ | 1 | 3 |
| 660 | 22 ⋅ 3 ⋅ 5 ⋅ 11 | A1 | 2 | 2 |
| 1092 | 22 ⋅ 3 ⋅ 7 ⋅ 13 | A1 | 2 | 2 |
| 2448 | 24 ⋅ 32 ⋅ 17 | A1 | 2 | 2 |
| 2520 | 23 ⋅ 32 ⋅ 5 ⋅ 7 | A7 | 6 | 2 |
| 3420 | 22 ⋅ 32 ⋅ 5 ⋅ 19 | A1 | 2 | 2 |
| 4080 | 24 ⋅ 3 ⋅ 5 ⋅ 17 | A1 | 1 | 4 |
| 5616 | 24 ⋅ 33 ⋅ 13 | A2 | 1 | 2 |
| 6048 | 25 ⋅ 33 ⋅ 7 | 2A2 = G2′ | 1 | 2 |
| 6072 | 23 ⋅ 3 ⋅ 11 ⋅ 23 | A1 | 2 | 2 |
| 7800 | 23 ⋅ 3 ⋅ 52 ⋅ 13 | A1 | 2 | 2×2 |
| 7920 | 24 ⋅ 32 ⋅ 5 ⋅ 11 | M11 | 1 | 1 |
| 9828 | 22 ⋅ 33 ⋅ 7 ⋅ 13 | A1 | 2 | 6 |
| 12180 | 22 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 29 | A1 | 2 | 2 |
| 14880 | 25 ⋅ 3 ⋅ 5 ⋅ 31 | A1 | 2 | 2 |
| 20160 | 26 ⋅ 32 ⋅ 5 ⋅ 7 | A3 = A8 | 2 | 2 |
| 20160 | 26 ⋅ 32 ⋅ 5 ⋅ 7 | A2 | 3×42 | D12 |
| 25308 | 22 ⋅ 32 ⋅ 19 ⋅ 37 | A1 | 2 | 2 |
| 25920 | 26 ⋅ 34 ⋅ 5 | 2A3 = B2 | 2 | 2 |
| 29120 | 26 ⋅ 5 ⋅ 7 ⋅ 13 | 2B2 | 22 | 3 |
| 32736 | 25 ⋅ 3 ⋅ 11 ⋅ 31 | A1 | 1 | 5 |
| 34440 | 23 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 41 | A1 | 2 | 2 |
| 39732 | 22 ⋅ 3 ⋅ 7 ⋅ 11 ⋅ 43 | A1 | 2 | 2 |
| 51888 | 24 ⋅ 3 ⋅ 23 ⋅ 47 | A1 | 2 | 2 |
| 58800 | 24 ⋅ 3 ⋅ 52 ⋅ 72 | A1 | 2 | 22 |
| 62400 | 26 ⋅ 3 ⋅ 52 ⋅ 13 | 2A2 | 1 | 4 |
| 74412 | 22 ⋅ 33 ⋅ 13 ⋅ 53 | A1 | 2 | 2 |
| 95040 | 26 ⋅ 33 ⋅ 5 ⋅ 11 | M12 | 2 | 2 |
lists the 56 non-cyclic simple groups of order less than a million.