List of finite simple groups

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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. (In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group Bn(q) has the same order as Cn(q) for q odd, n > 2; and the groups A8 = A3(2) and A2(4) both have orders 20160.)

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 (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram).

Infinite families[edit]

Cyclic groups Zp[edit]

Simplicity: Simple for p a prime number.

Order: p

Schur multiplier: Trivial.

Outer automorphism group: Cyclic of order p − 1.

Other names: Z/pZ

Remarks: These are the only simple groups that are not perfect.

An, n > 4, Alternating groups[edit]

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 (elementary abelian).

Other names: Altn.

There is an unfortunate conflict with the notation for the (unrelated) groups An(q), and 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(q) in italic.

Isomorphisms: A1 and A2 are trivial. A3 is cyclic of order 3. A4 is isomorphic to A1(3) (solvable). A5 is isomorphic to A1(4) and to A1(5). A6 is isomorphic to A1(9) and to the derived group B2(2)'. A8 is isomorphic to A3(2).

Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1.

Chevalley groups An(q), Bn(q) n > 1, Cn(q) n > 2, Dn(q) n > 3[edit]

Chevalley groups An(q)
linear groups
Chevalley groups Bn(q) n > 1
orthogonal groups
Chevalley groups Cn(q) n > 2
symplectic groups
Chevalley groups Dn(q) n > 3
orthogonal groups
Simplicity: A1(2) and A1(3) are solvable, the others are simple. B2(2) is not simple but its derived group B2(2)′ is a simple subgroup of index 2; the others are simple. All simple All simple
Order: \frac{q^{\frac{1}{2} n(n+1)}}{(n+1,q-1)} \prod_{i=1}^n(q^{i+1}-1) \frac{q^{n^2}}{(2,q-1)}\prod_{i=1}^n(q^{2i}-1) \frac{q^{n^2}}{(2,q-1)}\prod_{i=1}^n(q^{2i}-1) \frac{q^{n(n-1)}(q^n-1)}{(4,q^n-1)}\prod_{i=1}^{n-1}(q^{2i}-1)
Schur multiplier: For the simple groups it is cyclic of order (n+1, q − 1) except for A1(4) (order 2), A1(9) (order 6), A2(2) (order 2), A2(4) (order 48, product of cyclic groups of orders 3, 4, 4), A3(2) (order 2). (2,q − 1) except for B2(2) = S6 (order 2 for B2(2), order 6 for B2(2)′) and B3(2) (order 2) and B3(3) (order 6). (2,q − 1) except for C3(2) (order 2). The order is (4, qn − 1) (cyclic for n odd, elementary abelian for n even) except for D4(2) (order 4, elementary abelian).
Outer automorphism group: (2, q − 1) ·f·1 for n = 1; (n+1, q − 1) ·f·2 for n > 1, where q = pf. (2, q − 1) ·f·1 for q odd or n>2; (2, q − 1) ·f·2 if q is even and n=2, where q = pf. (2, q − 1) ·f·1 where q = pf. (2, q − 1) 2·f·S3 for n=4, (2, q − 1) 2·f·2 for n>4 even, (4, qn − 1)·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(q), Ln+1(q), PSL(n+1,q) O2n+1(q), Ω2n+1(q) (for q odd). Projective symplectic group, PSp2n(q), PSpn(q) (not recommended), S2n(q), Abelian group (archaic). O2n+(q), 2n+(q). "Hypoabelian group" is an archaic name for this group in characteristic 2.
Isomorphisms: A1(2) is isomorphic to the symmetric group on 3 points of order 6. A1(3) is isomorphic to the alternating group A4 (solvable). A1(4) and A1(5) are isomorphic, and are both isomorphic to the alternating group A5. A1(7) and A2(2) are isomorphic. A1(8) is isomorphic to the derived group 2G2(3)′. A1(9) is isomorphic to A6 and to the derived group B2(2)′. A3(2) is isomorphic to A8. Bn(2m) is isomorphic to Cn(2m). B2(2) is isomorphic to the symmetric group on 6 points, and the derived group B2(2)′ is isomorphic to A1(9) and to A6. B2(3) is isomorphic to 2A3(22). Cn(2m) is isomorphic to Bn(2m)
Remarks: These groups are obtained from the general linear groups GLn+1(q) by taking the elements of determinant 1 (giving the special linear groups SLn+1(q)) 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(q) also exists, but is the same as A1(q). B2(q) 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(q) also exists, but is the same as A1(q). C2(q) also exists, but is the same as B2(q). This is the group obtained from the split orthogonal group in dimension 2n by taking the kernel of the determinant (or Dickson invariant in characteristic 2) 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(q) also exists, but is the same as A1(qA1(q). D3(q) also exists, but is the same as A3(q).

Chevalley groups E6(q), E7(q), E8(q), F4(q), G2(q)[edit]

Chevalley groups E6(q) Chevalley groups E7(q) Chevalley groups E8(q) Chevalley groups F4(q) Chevalley groups G2(q)
Simplicity: All simple All simple All simple All simple G2(2) is not simple but its derived group G2(2)′ is a simple subgroup of index 2; the others are simple.
Order: q36(q12 − 1)(q9 − 1)(q8 − 1)(q6 − 1)(q5 − 1)(q2 − 1)/(3,q − 1) q63(q18 − 1)(q14 − 1)(q12 − 1)(q10 − 1)(q8 − 1)(q6 − 1)(q2 − 1)/(2,q − 1) q120(q30−1)(q24−1)(q20−1)(q18−1)(q14−1)(q12−1)(q8−1)(q2−1) q24(q12−1)(q8−1)(q6−1)(q2−1) q6(q6−1)(q2−1)
Schur multiplier: (3,q − 1) (2,q − 1) Trivial Trivial except for F4(2) (order 2). Trivial for the simple groups except for G2(3) (order 3) and G2(4) (order 2).
Outer automorphism group: (3, q − 1) ·f·2 where q = pf. (2, q − 1) ·f·1 where q = pf. f·1 where q = pf. f·1 for q odd, 1·f·2 for q even, where q = pf. 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(2)′ is isomorphic to 2A2(32).
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(3) 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(q) 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(q) 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 groups 2An(q2) n > 1, 2Dn(q2) n > 3, 2E6(q2), 3D4(q3)[edit]

Steinberg groups 2An(q2) n > 1
unitary groups
Steinberg groups 2Dn(q2) n > 3
orthogonal groups
Steinberg groups 2E6(q2) Steinberg groups 3D4(q3)
Simplicity: 2A2(22) is solvable, the others are simple. All simple All simple All simple
Order: {1\over (n+1,q+1)}q^{n(n+1)/2}\prod_{i=1}^n(q^{i+1}-(-1)^{i+1}) {1\over (4,q^n+1)}q^{n(n-1)}(q^n+1)\prod_{i=1}^{n-1}(q^{2i}-1) q36(q12−1)(q9+1)(q8−1)(q6−1)(q5+1)(q2−1)/(3,q+1) q12(q8+q4+1)(q6−1)(q2−1)
Schur multiplier: Cyclic of order (n + 1, q + 1) for the simple groups, except for 2A3(22) (order 2), 2A3(32) (order 36, product of cyclic groups of orders 3,3,4), 2A5(22) (order 12, product of cyclic groups of orders 2,2,3) Cyclic of order (4, qn + 1) (3, q + 1) except for 2E6(22) (order 12, product of cyclic groups of orders 2,2,3). Trivial
Outer automorphism group: (n+1, q + 1) ·f·1 where q2 = pf (4, qn + 1) ·f·1 where q2 = pf (3, q + 1) ·f·1 where q2 = pf. f·1 where q3 = pf.
Other names: Twisted Chevalley group, projective special unitary group, PSUn+1(q), PSU(n+1, q), Un+1(q), 2 An(q), 2An(q, q2) 2Dn(q), O2n(q), 2n(q), twisted Chevalley group. "Hypoabelian group" is an archaic name for this group in characteristic 2. 2E6(q), twisted Chevalley group. 3D4(q), D42(q3), Twisted Chevalley groups.
Isomorphisms: The solvable group 2A2(22) is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. 2A2(32) is isomorphic to the derived group G2(2)′. 2A3(22) is isomorphic to B2(3).
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 (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. 2D2(q2) also exists, but is the same as A1(q2). 2D3(q2) also exists, but is the same as 2A3(q2). One of the exceptional double covers of 2E6(22) 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(23) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.

2B2(22n+1) Suzuki groups[edit]

Simplicity: Simple for n ≥ 1. The group 2B2(2) is solvable.

Order: q2 (q2 + 1) (q − 1) where q = 22n+1.

Schur multiplier: Trivial for n ≠ 1, elementary abelian of order 4 for 2B2(8).

Outer automorphism group:

f·1

where f = 2n + 1.

Other names: Suz(22n+1), Sz(22n+1).

Isomorphisms: 2B2(2) is the Frobenius group of order 20.

Remarks: Suzuki group are Zassenhaus groups acting on sets of size (22n+1)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.

2F4(22n+1) Ree groups, Tits group[edit]

Simplicity: Simple for n ≥ 1. The derived group 2F4(2)′ is simple of index 2 in 2F4(2), and is called the Tits group, named for the Belgian mathematician Jacques Tits.

Order: q12 (q6 + 1) (q4 − 1) (q3 + 1) (q − 1) 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:

f·1

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.

2G2(32n+1) Ree groups[edit]

Simplicity: Simple for n ≥ 1. The group 2G2(3) is not simple, but its derived group 2G2(3)′ is a simple subgroup of index 3.

Order: q3 (q3 + 1) (q − 1) where q = 32n+1

Schur multiplier: Trivial for n≥1 and for 2G2(3)′.

Outer automorphism group:

f·1

where f = 2n + 1.

Other names: Ree(32n+1), R(32n+1), E2*(32n+1) .

Isomorphisms: The derived group 2G2(3)′ is isomorphic to A1(8).

Remarks: 2G2(32n+1) has a doubly transitive permutation representation on 33(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 32n+1 elements.

Sporadic groups[edit]

Mathieu groups M11, M12, M22, M23, M24[edit]

Mathieu group M11 Mathieu group M12 Mathieu group M22 Mathieu group M23 Mathieu group M24
Order: 24 · 32 · 5 · 11=7920 26 · 33 · 5 · 11=95040 27 · 32 · 5 · 7 · 11 = 443520 27 · 32 · 5 · 7 · 11 · 23=10200960 210 · 33 · 5 · 7 · 11 · 23= 244823040
Schur multiplier: Trivial Order 2 Cyclic of order 12[a] Trivial Trivial
Outer automorphism group: Trivial Order 2 Order 2 Trivial Trivial
Remarks: A 4-transitive permutation group on 11 points, and the point stabilizer in M12. The subgroup fixing a point is sometimes called M10, and has a subgroup of index 2 isomorphic to the alternating group A6. A 5-transitive permutation group on 12 points. A 3-transitive permutation group on 22 points. A 4-transitive permutation group on 23 points, contained in M24. A 5-transitive permutation group on 24 points.

Janko groups J1, J2, J3, J4[edit]

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(1), J(11) Hall–Janko group, HJ
Remarks: It is a subgroup of G2(11), and so has a 7 dimensional representation over the field with 11 elements. It is the automorphism group of a rank 3 graph on 100 points called the Hall-Janko graph, and is also contained in G2(4). J3 seems unrelated to any other sporadic groups (or to anything else). 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 groups Co1, Co2, Co3[edit]

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 of Co1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0. Subgroup of Co1; fixes a norm 4 vector in the Leech lattice. Subgroup of Co1; fixes a norm 6 vector in the Leech lattice. It has a doubly transitive permutation representation on 276 points.

Fischer groups Fi22, Fi23, Fi24'[edit]

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(22) M(23) M(24)′, 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[edit]

Order: 29 · 32 · 53· 7 · 11 = 44352000

Schur 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 Co3.

McLaughlin group McL[edit]

Order: 27 · 36 · 53· 7 · 11 = 898128000

Schur 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 Co3.

Held group He[edit]

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[edit]

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[edit]

Order: 213 · 37 · 52· 7 · 11 · 13 = 448345497600

Schur 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[edit]

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[edit]

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[edit]

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 Th[edit]

Order: 215 · 310 · 53 · 72 · 13 · 19 · 31 = 90745943887872000

Schur 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(3), so has a 248-dimensional representation over the field with 3 elements.

Baby Monster group B[edit]

Order:

   241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 47
= 4154781481226426191177580544000000

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 (with no nontrivial invariant product), and a representation of dimension 4370 over the field with 2 elements preserving a commutative but non-associative product.

Fischer–Griess Monster group M[edit]

Order:

   246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
= 808017424794512875886459904961710757005754368000000000

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 196884 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[edit]

Order Factored order Group Schur multiplier Outer automorphism group
60 22 · 3 · 5 A5 = A1(4) = A1(5) 2 2
168 23 · 3 · 7 A1(7) = A2(2) 2 2
360 23 · 32 · 5 A6 = A1(9) = B2(2)′ 6 2×2
504 23 · 32 · 7 A1(8) = 2G2(3)′ 1 3
660 22 · 3 · 5 · 11 A1(11) 2 2
1092 22 · 3 · 7 · 13 A1(13) 2 2
2448 24 · 32 · 17 A1(17) 2 2
2520 23 · 32 · 5 · 7 A7 6 2
3420 22 · 32 · 5 · 19 A1(19) 2 2
4080 24 · 3 · 5 · 17 A1(16) 1 4
5616 24 · 33 · 13 A2(3) 1 2
6048 25 · 33 · 7 2A2(9) = G2(2)′ 1 2
6072 23 · 3 · 11 · 23 A1(23) 2 2
7800 23 · 3 · 52 · 13 A1(25) 2 2×2
7920 24 · 32 · 5 · 11 M11 1 1
9828 22 · 33 · 7 · 13 A1(27) 2 6
12180 22 · 3 · 5 · 7 · 29 A1(29) 2 2
14880 25 · 3 · 5 · 31 A1(31) 2 2
20160 26 · 32 · 5 · 7 A3(2) = A8 2 2
20160 26 · 32 · 5 · 7 A2(4) 3×42 D12
25308 22 · 32 · 19 · 37 A1(37) 2 2
25920 26 · 34 · 5 2A3(4) = B2(3) 2 2
29120 26 · 5 · 7 · 13 2B2(8) 22 3
32736 25 · 3 · 11 · 31 A1(32) 1 5
34440 23 · 3 · 5 · 7 · 41 A1(41) 2 2
39732 22 · 3 · 7 · 11 · 43 A1(43) 2 2
51888 24 · 3 · 23 · 47 A1(47) 2 2
58800 24 · 3 · 52 · 72 A1(49) 2 22
62400 26 · 3 · 52 · 13 2A2(16) 1 4
74412 22 · 33 · 13 · 53 A1(53) 2 2
95040 26 · 33 · 5 · 11 M12 2 2

(Complete for orders less than 100,000)

Hall (1972) lists the 56 non-cyclic simple groups of order less than a million.

See also[edit]

Notes[edit]

  1. ^ There were several mistakes made in the initial calculations of the Schur multiplier, so some older books and papers list incorrect values. (This caused an error in the title of Janko's original 1976 paper[1] giving evidence for the existence of the group J4. At the time it was thought that the full covering group of M22 was 6·M22. In fact J4 has no subgroup 12·M22.)

References[edit]

  1. ^ "A new finite simple group of order 86,775,571,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups.". J. Algebra 42: 564–596. 1976. doi:10.1016/0021-8693(76)90115-0. 

Further reading[edit]

External links[edit]