The full automorphism groups of the five symmetric $(15,8,4)$-designs

TL;DR

This paper determines the full automorphism groups of four symmetric (15,8,4)-designs using geometric methods, complementing the known automorphism group of the design derived from projective geometry.

Contribution

It extends the understanding of automorphism groups to four additional symmetric (15,8,4)-designs using point-line geometry techniques.

Findings

Identified the full automorphism groups of four symmetric (15,8,4)-designs.

Described the actions of these groups on points and blocks.

Complemented the known automorphism group of the PG(3,2) design.

Abstract

It is clear that the full automorphism group of the $(15,8,4)$-design of points and hyperplane complements of ${\rm PG}(3,2)$ is ${\rm GL}(4,2)$. Using methods of point-line geometries, we determine the full automorphism groups of the remaining four symmetric $(15,8,4)$-designs and describe their actions on the sets of points and blocks.