New 3-designs and 2-designs having U(3,3) as an automorphism group

In this paper we classify 2-designs and 3-designs with 28 or 36 points admitting a transitive action of the unitary group U(3,3) on points and blocks. We also construct 2-designs and 3-designs with 56 or 63 points and strongly regular graphs on 36, 63 or 126 vertices having U(3,3) as a transitive automorphism group. Further, we show that this completes the classification of 3-designs admitting a transitive action of the group U(3,3), in terms of parameters. A number of the 3-designs and 2-designs obtained in this paper have not been known before up to our best knowledge.