In this article, we study symmetric $(v, k, λ)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is $PSU_{4}(q)$. We prove that there exist eight non-isomorphic such designs for which $λ\in\{3,6,18\}$ and $G$ is either $PSU_{4}(2)$, or $PSU_{4}(2):2$.
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