On Designs Constructed by Group Actions

Abstract

Using the action of the linear fractional groups $L_2(q)$, where $q=8,25,27,29,31$ and $32$, some $1$-designs are constructed. It is shown that subgroups of the automorphism group of $L_2(q)$ appear as the full automorphism group of the constructed designs. In the cases $q=8$ and $32$, it is shown that the symmetric groups ${\mathbb{S}}_9$ and ${\mathbb{S}}_{33}$, respectively, appear as the automorphism group of one of the constructed designs.