Block-transitive 3-(v, k, 1) designs on exceptional groups of Lie type

Let \({\mathcal {D}}\) $\mathcal{D}$ be a non-trivial G-block-transitive 3-(v, k, 1) design, where \(T\le G \le \textrm{Aut}(T)\) $T\le G\le \text{Aut}(T)$ for some finite non-abelian simple group T. It is proved that if T is a simple exceptional group of Lie type, then T is either the Suzuki group \({}^2B_2(q)\) ${}^2{B}_2(q)$ or \(G_2(q)\) $G_2(q)$ . Furthermore, if \(T={}^2B_2(q)\) $T={}^2{B}_2(q)$ then the design \({\mathcal {D}}\) $\mathcal{D}$ has parameters \(v=q^2+1\) $v=q^2+1$ and \(k=q+1\) $k=q+1$ , and so \({\mathcal {D}}\) $\mathcal{D}$ is an inverse plane of order q, and if \(T=G_2(q)\) $T=G_2(q)$ then the point stabilizer in T is either \(\textrm{SL}_3(q).2\) ${\text{SL}}_3(q).2$ or \(\textrm{SU}_3(q).2\) ${\text{SU}}_3(q).2$ , and the parameter k satisfies very restricted conditions.