Steinberg triality groups acting on 2-(v,k,1)designs

A 2 - (v, k, 1) design D = (P,B) is a system consisting of a finite set P of v points and a collection B of k-subsets of P, called blocks, such that each 2-subset of P is contained in precisely one block.Let G be an automorphism group of a 2 - (v, k, 1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ≤ G ≤ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to 3D4(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.

A 2 - (υ, k, 1) design D = (?,?, ?) is a system consisting of a finite set ? of υ points and a collection ? of ?-subsets of ?, called blocks, such that each 2-subset of ? is contained in precisely one block. Let G be an automorphism group of a 2-(υ, k, 1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ? G ? Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to 3 D 4(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.