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

Abstract

It is clear that the full automorphism group of the $(15,8,4)$-design of points and hyperplane complements of $\mathrm{PG}(3,2)$ is $\mathrm{GL}(4,2)$. 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.