Block-Transitive Steiner 3-Designs on 2-Transitive Groups

In 2005, Huber proved that the flag-transitive group of automorphisms for a Steiner [Formula: see text]-design must act [Formula: see text]-transitively on the point set. He also classified flag-transitive Steiner [Formula: see text]-designs. It should be noted that while flag-transitive designs are block-transitive, the converse is not always true. Furthermore, all known block-transitive Steiner [Formula: see text]-designs have [Formula: see text]-transitive automorphism groups. In this study, we consider [Formula: see text]-block-transitive Steiner [Formula: see text]-designs [Formula: see text] with [Formula: see text] acting [Formula: see text]-transitively on the point set, and prove that if [Formula: see text] is not [Formula: see text]-flag-transitive, then either [Formula: see text] is a [Formula: see text]-[Formula: see text]-design with the Suzuki group as its automorphism group, or [Formula: see text] is a subgroup of [Formula: see text] and the size of the block is an odd integer.