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, E6(q), E7(q), E8(q), F4(q), G2(q) Steinberg groups, 2An(q2) n > 1, 2Dn(q2) n > 3, 2E6(q2), 3D4(q3) 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). 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. Ree groups and Tits group, 2F4(22n+1) 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