Constructing flag-transitive, point-imprimitive designs

TL;DR

This paper introduces a new construction method for flag-transitive, point-imprimitive designs, characterizes their automorphism groups, and provides explicit examples including a novel symmetric design and symplectic designs with specific automorphism groups.

Contribution

It presents a construction framework for such designs, determines their automorphism subgroups, and offers concrete examples including a new symmetric design and symplectic designs.

Findings

A new symmetric 2-(1408,336,80) design with a specific automorphism group.

Conditions for designs to have flag-transitive automorphism groups.

Construction of symplectic designs with flag-transitive, point-imprimitive automorphism groups.

Abstract

We give a construction of a family of designs with a specified point-partition, and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric $2$-$(1408,336,80)$ design with automorphism group $2^{12}:((3\cdot\mathrm{M}_{22}):2)$, and a construction of one of the families of the symplectic designs (the designs $S^-(n)$) exhibiting a flag-transitive, point-imprimitive automorphism group.