The $2-(v,13,1)$ Designs with Block Transitive Automorphism

Abstract

This article is a contribution to the study of the automorphism groups
of $2$-$(v,k,1)$ designs. Let $\mathcal{D}$ be a $2$-$(v,13,1)$ design and
suppose that $G$ is a group of automorphisms of $\mathcal{D}$ which is
block-transitive and point-primitive. Then $\mathrm{S}\mathrm{o}\mathrm{c}(G)$,
the socle of $G$, is not isomorphic to ${}^2{G}_2(q)$ or to ${}^2{F}_4(q^2)$
for any prime power $q$.