Symmetric designs and four dimensional projective special unitary groups

In this article, we study symmetric $(v,k,\lambda )$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is $PS{U}_4(q)$. We prove that there exist eight non-isomorphic such designs for which $\lambda \in \{3,6,18\}$ and $G$ is either $PS{U}_4(2)$, or $PS{U}_4(2):2$.