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.