# 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.

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.

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).

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.

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

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.

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.

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.

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.

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.

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.

Remarks: Acts as a rank 3 permutation group on the McLaughlin graph with 275 points, and is contained in Co2 and in Co3.

Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers.

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.

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

Remarks: Has a 111-dimensional representation over the field with 5 elements.

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.

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.

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.

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