Automorphism Groups of Finite Simple Groups

AutomorphismGroupSimpleGroup(type, d, q): MonStgElt, RngIntElt, RngIntElt -> GrpPerm

AutomorphismGroupSimpleGroup(type, q): MonStgElt, RngIntElt -> GrpPerm

AutomorphismGroupSimpleGroup(name): MonStgElt -> GrpPerm

AutomorphismGroupSimpleGroup(triple): Tup -> GrpPerm

Compute and return a permutation group isomorphic to the full automorphism group of the finite simple group specified by the input parameters.

The first form is for classical groups. type can be “L”, “U”, “S”, “O” (for odd dimensions), “O+”, or “O-”, d is the dimension, and q is the order of the field of definition.

The second form is for alternating or cyclic groups, or exceptional groups of Lie type. type can be “A” (for alternating), “C” (for cyclic), “G2”, “Sz” (“Suz”), “2B2” (“TB2”), “Ree” (“R”), “2G2” (“TG2”), “3D4” (“TD4”), “F4”, “E6”, or “2F4” (“TF4”). The types in brackets are alternatives. For the missing types “2E6”, “E7” and “E8”, there are no permutation representations of sufficiently small degree. The second input parameter q is the order of the field of definitions except for types “A” or “C”, when it is the degree.

The third form is for sporadic groups. type can be “M11”, “M12”, “M22”, “M23”, “M24”, “J1”, “HS”, “J2”, “HJ”, “McL”, “Suz”, “J3”, “Co1”, “Co2”, “Co3”, “He”, “Fi22” (“F22”), “Fi23” (“F23”), “Fi24” (“F24”),”Ru”, “ON”, or “HN”.

In the fourth form, the input is the triple of integers \(<i,j,k>\), which is used by Magma to specify a finite simple group. Such a triple is returned as the second return value of IsSimpleOrder for example.