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 notation (a,b) represents the greatest common divisor of the integers a and b.
Outer automorphism group: Cyclic of order p − 1.
Other names: Z/pZ, Cp Remarks: These are the only simple groups that are not perfect.
Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1.
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 outer automorphism group is often, but not always, isomorphic to the semidirect product
are cyclic of the respective orders d, f, g, except for type
The notation (a,b) represents the greatest common divisor of the integers a and b.
Schur multiplier: Trivial for n ≠ 1, elementary abelian of order 4 for 2B2(8).
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: 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.
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.
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.
Other names: F5, D Remarks: Centralizes an element of order 5 in the monster group.
Other names: Lyons–Sims group, LyS Remarks: Has a 111-dimensional representation over the field with 5 elements.
Other names: F3, E Remarks: Centralizes an element of order 3 in the monster.
Has a 248-dimensional representation which, when reduced modulo 3, leads to containment in E8(3).
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.
Remarks: Contains all but 6 of the other sporadic groups as subquotients.