Geometric groups of second order and related combinatorial structures

In 1977, D. Betten defined a geometric group to be a permutation group ( G , Ω ) such that G = Aut ( R ) for some hypergraph R on Ω . In this paper, we extend Betten's notion of a geometric group to what we call a geometric group of second order. By definition, this is a permutation group for which G = Aut ( R ) for some set R = { R 1 , R 2 , … , R d } of hypergraphs on Ω . Our main focus will be on permutation groups that are geometric of second order but not geometric. Within this small class of groups one finds the projective groups P G L ( 2 , 8 ) , P Γ L ( 2 , 8 )and the affine groups A G L ( 1 , 8 ) , A Γ L ( 1 , 8 ) . Our investigations, which are based primarily on these four groups, lead us to consider some familiar combinatorial structures (eg, Fano plane and affine design) in a less familiar context (overlarge sets of Steiner systems).