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# 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 (including the Tits group, which strictly speaking is not 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).

 Contents

## Infinite families

### Cyclic groupsZp

Simplicity: Always simple.

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 > 1, Alternating groups

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.

Outer automorphism group: 2, except for n=1, n=2 when it it trivial, and n=6 when it has order 4 (elementary abelian).

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.

### An(q) Chevalley groups, linear groups

Simplicity: A1(2) and A1(3) are solvable, the others are simple.

Order:

${1\over (n+1,q-1)} q^{n(n+1)/2} \prod_{i=1}^n(q^{i+1}-1)$

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).

Outer automorphism group: (2, q − 1)·f·1 for n=1; (n+1, q − 1)·f·2 for n>1, where q = pf.

Other names: Projective special linear groups, PSLn+1(q), Ln+1(q), PSL(n+1,q)

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.

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.

### Bn(q) n > 1 Chevalley groups, orthogonal group

Simplicity: B2(2) is not simple and has a simple subgroup of index 2; the others are simple.

Order:

${1\over (2,q-1)} q^{n^2} \prod_{i=1}^n(q^{2i}-1)$

Schur multiplier: (2,q − 1) except for B2(2) (not simple) and B3(2) (order 2) and B3(3) (order 6).

Outer automorphism group: (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.

Other names: O2n+1(q), Ω2n+1(q) (for q odd).

Isomorphisms: 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).

Remarks: 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.

### Cn(q) n > 2 Chevalley groups, symplectic groups

Simplicity: All simple.

Order:

${1\over (2,q-1)} q^{n^2} \prod_{i=1}^n(q^{2i}-1)$

Schur multiplier: (2,q − 1) except for C3(2) (order 2).

Outer automorphism group: (2, q − 1)·f·1 where q = pf.

Other names: Projective symplectic group, PSp2n(q), PSpn(q) (not recommended), S2n(q).

Isomorphisms: Cn(2m) is isomorphic to Bn(2m)

Remarks: 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).

### Dn(q) n > 3 Chevalley groups, orthogonal groups

Simplicity: All simple.

Order:

${1\over (4,q^n-1)} q^{n(n-1)} (q^n-1) \prod_{i=1}^{n-1}(q^{2i}-1)$

Schur multiplier: 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)2·f·S3 for n=4, (2, q − 1)2·f·2 for n>4 even, (4, qn − 1)2·f·2 for n odd, where q = pf, and S3 is the symmetric group on 3 points of order 6.

Other names: O2n+(q), 2n+(q).

Remarks: 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(q) also exists, but is the same as A1(qA1(q). D3(q) also exists, but is the same as A3(q).

### E6(q) Chevalley groups

Simplicity: All simple.

Order: q36 (q12−1) (q9−1) (q8−1) (q6−1) (q5−1) (q2−1) /(3,q-1)

Schur multiplier: (3,q − 1)

Outer automorphism group: (3, q − 1)·f·2 where q = pf.

Other names: Exceptional Chevalley group.

Remarks: Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.

### E7(q) Chevalley groups

Simplicity: All simple.

Order: q63 (q18−1) (q14−1) (q12−1) (q10−1) (q8−1) (q6−1) (q2−1) /(2,q-1)

Schur multiplier: (2,q − 1)

Outer automorphism group: (2, q − 1)·f·1 where q = pf.

Other names: Exceptional Chevalley group.

Remarks: Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.

### E8(q) Chevalley groups

Simplicity: All simple.

Order: q120 (q30−1) (q24−1) (q20−1) (q18−1) (q14−1) (q12−1) (q8−1) (q2−1)

Schur multiplier: Trivial.

Outer automorphism group:f·1 where q = pf.

Other names: Exceptional Chevalley group.

Remarks: It acts on the corresponding Lie algebra of dimension 248. E8(3) contains the Thompson simple group.

### F4(q) Chevalley groups

Simplicity: All simple.

Order: q24 (q12−1) (q8−1) (q6−1) (q2−1)

Schur multiplier: Trivial except for F4(2) (order 2).

Outer automorphism group:f·1 for q odd, 1·f·2 for q even, where q = pf.

Other names: Exceptional Chevalley group.

Remarks: 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.

### G2(q) Chevalley groups

Simplicity: G2(2) is not simple but has a simple subgroup of index 2; the others are simple.

Order: q6 (q6−1) (q2−1)

Schur multiplier: Trivial for the simple groups except for G2(3) (order 3) and G2(4) (order 2).

Outer automorphism group: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.

Isomorphisms: The derived group G2(2)′ is isomorphic to 2A2(32).

Remarks: 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.

### 2An(q2) n > 1 Steinberg groups, unitary groups

Simplicity: 2A2(22) is solvable, the others are simple.

Order:

${1\over (n+1,q+1)} q^{n(n+1)/2} \prod_{i=1}^n(q^{i+1}-(-1)^{i+1})$

Schur multiplier: (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)

Outer automorphism group: (n+1, q + 1)·f·1 where q2 = pf.

Other names: Twisted chevalley group, projective special unitary group, PSUn+1(q), PSU(n+1, q), Un+1(q), 2An(q), 2An(q, q2)

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.

### 2Dn(q2) n > 3 Steinberg groups, orthogonal groups

Simplicity: All simple.

Order:

${1\over (4,q^n+1)} q^{n(n-1)} (q^n+1) \prod_{i=1}^{n-1}(q^{2i}-1)$

Schur multiplier: Cyclic of order (4, qn + 1).

Outer automorphism group: (4, qn + 1)·f·1 where q2 = pf.

Other names: 2Dn(q), O2n(q), 2n(q), twisted chevalley group.

Remarks: 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(q2) also exists, but is the same as A1(q). 2D3(q2) also exists, but is the same as 2A3(q2).

### 2E6(q2) Steinberg groups

Simplicity: All simple.

Order: q36 (q12−1) (q9+1) (q8−1) (q6−1) (q5+1) (q2−1) /(3,q+1)

Schur multiplier: (3, q + 1) except for 2E6(22) (order 12, product of cyclic groups of orders 2,2,3).

Outer automorphism group: (3, q + 1)·f·1 where q2 = pf.

Other names: 2E6(q), twisted Chevalley group.

Remarks: 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(q3) Steinberg groups

Simplicity: All simple.

Order: q12 (q8+q4+1) (q6−1) (q2−1)

Schur multiplier: Trivial.

Outer automorphism group:f·1 where q3 = pf.

Other names: 3D4(q), Twisted Chevalley groups.

Remarks: 3D4(23) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.

### 2B2(22n+1) Suzuki groups

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>2, 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, 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

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: The Tits group is strictly speaking not a group of Lie type, and in particular it is not the group of points of a connected simple algebraic group with values in some field, nor does it have a BN pair. However most authors count it as a sort of honorary group of Lie type.

### 2G2(32n+1) Ree groups

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.

Outer automorphism group:f·1 where f = 2n+1.

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

### Mathieu groupM11

Order: 24 · 32 · 5 · 11=7920

Schur multiplier: Trivial.

Outer automorphism group: 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.

### Mathieu groupM12

Order: 26 · 33 · 5 · 11=95040

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

Remarks: A 5-transitive permutation group on 12 points.

### Mathieu groupM22

Order: 27 · 32 · 5 · 7 · 11=443520

Schur multiplier: Cyclic of order 12. There were several mistakes made in the initial calculations of the Schur multiplier, so some older books and papers list incorrect values.

Outer automorphism group: Order 2.

Remarks: A 3-transitive permutation group on 22 points.

### Mathieu groupM23

Order: 27 · 32 · 5 · 7 · 11 · 23=10200960

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Remarks: A 4-transitive permutation group on 23 points, contained in M24.

### Mathieu groupM24

Order: 210 · 33 · 5 · 7 · 11 · 23= 244823040

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Remarks: A 5-transitive permutation group on 24 points.

### Janko groupJ1

Order: 23 · 3 · 5 · 7 · 11 · 19 = 175560

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: J(1), J(11)

Remarks: It is a subgroup of G2(11), and so has a 7 dimensional representation over the field with 11 elements.

### Janko groupJ2

Order: 27 · 33 · 52 · 7 = 604800

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

Other names: Hall-Janko group, HJ

Remarks: It is the automorphism group of a rank 3 graph on 100 points, and is also contained in G2(4).

### Janko groupJ3

Order: 27 · 35 · 5 · 17 · 19 = 50232960

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: Higman-Janko-McKay group, HJM

Remarks: 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.

### Janko groupJ4

Order: 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Remarks: Has a 112 dimensional representation over the field with 2 elements.

### Conway groupCo1

Order: 221 · 39 · 54 · 72 · 11 · 13 · 23 = 4157776806543360000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Other names: ·1

Remarks: The perfect double cover of Co1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0.

### Conway groupCo2

Order: 218 · 36 · 53 · 7 · 11 · 23 = 42305421312000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: ·2

Remarks: Subgroup of Co1; fixes a norm 4 vector in the Leech lattice.

### Conway groupCo3

Order: 210 · 37 · 53 · 7 · 11 · 23 = 495766656000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: ·3

Remarks: Subgroup of Co1; fixes a norm 6 vector in the Leech lattice.

### Fischer groupFi22

Order: 217 · 39 · 52 · 7 · 11 · 13 = 64561751654400.

Schur multiplier: Order 6.

Outer automorphism group: Order 2.

Other names: M(22)

Remarks: A 3-transposition group whose double cover is contained in Fi23.

### Fischer groupFi23

Order: 218 · 313 · 52 · 7 · 11 · 13 · 17 · 23 = 4089470473293004800.

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: M(23)

Remarks: A 3-transposition group contained in Fi24.

### Fischer groupFi24

Order: 221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29 = 1255205709190661721292800.

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: M(24)′, Fi24′.

Remarks: The triple cover is contained in the monster group.

### Higman-Sims groupHS

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 groupMcL

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 McLauglin graph with 275 points, and is contained in Co3.

### Held groupHe

Order: 210 · 33 · 52· 73· 17 = 4030387200

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

Other names: Held-Higman-McKay group, HHM, F7

Remarks: Centralizes an element of order 7 in the monster group.

### Rudvalis groupRu

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.

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 groupO'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

Remarks: The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.

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 groupLy

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 groupTh

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).

### Baby Monster groupB

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.

### Fischer-Griess Monster groupM

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.