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. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = A3(2) and A2(4) both have order 20160, and that the group Bn(q) has the same order as Cn(q) for q odd, n > 2. The smallest of the latter pairs of groups are B3(3) and C3(3) which both have order 4585351680.)
There is an unfortunate conflict between the notations for the alternating groups An and the groups of Lie type An(q). 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.
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 (a,b) represents the greatest common divisor of the integers a and b.
Class Family Order Exclusions Duplicates Cyclic groups Zp p prime Alternating groups Ann > 4 Classical Chevalley groups An(q) A1(2), A1(3) Bn(q)n > 1 B2(2) Cn(q)n > 2 Cn(2m) ≃ Bn(2m) Dn(q)n > 3 Exceptional Chevalley groups E6(q) E7(q) E8(q) F4(q) G2(q) G2(2) Classical Steinberg groups 2An(q2)n > 1 2A2(22) 2A3(22) ≃ B2(3) 2Dn(q2)n > 3 Exceptional Steinberg groups 2E6(q2) 3D4(q3) Suzuki groups 2B2(q)q = 22n+1 Ree groups+ Tits group 2F4(q)q = 22n+1 2F4(2)′ 212(26 + 1)(24 − 1)(23 + 1)(2 − 1)/2 = 2G2(q)q = 32n+1 Mathieu groups M11 M12 M22 M23 M24 Janko groups J1 J2 J3 J4 Conway groups Co3 Co2 Co1 Fischer groups Fi22 Fi23 Fi24′ Higman–Sims group HS McLaughlin group McL Held group He Rudvalis group Ru Suzuki sporadic group Suz O'Nan group O'N Harada–Norton group HN Lyons group Ly Thompson group Th Baby Monster group B Monster group M
Cyclic groups, Zp 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 groups, An, n = 3, n > 4
Simplicity: Solvable for n ≤ 2 and n = 4, 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.
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.
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 (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). The outer automorphism group is often, but not always, 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 (only when ) , the symmetric group on three elements. The notation (a,b) represents the greatest common divisor of the integers a and b.
Chevalley groups, An(q), Bn(q) n > 1, Cn(q) n > 2, Dn(q) n > 3
Chevalley groups, An(q) linear groups Chevalley groups, Bn(q) n > 1orthogonal groups Chevalley groups, Cn(q) n > 2symplectic groups Chevalley groups, Dn(q) n > 3orthogonal 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 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 for q 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), PΩ2n+(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 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(q)×A1(q). D3(q) also exists, but is the same as A3(q).
Chevalley groups, E6(q), E7(q), E8(q), F4(q), G2(q)
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 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(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)
Steinberg groups, 2An(q2) n > 1unitary groups Steinberg groups, 2Dn(q2) n > 3orthogonal 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 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 1⋅f⋅1, where q3 = 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) 2Dn(q), O2n−(q), PΩ2n−(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.
Suzuki groups, 2B2(22n+1)
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: 1⋅f⋅1, where f = 2n + 1.
Other names: Suz(22n+1), Sz(22n+1).