symmetric designs and projective special unitary groups psu5(q)

In this article, we prove that if a nontrivial symmetric (v, k, λ) design admit a flag-transitive and point-primitive automorphism group g, then the socle x of g cannot be a projective special unitary group of dimension five. as a corollary, we list all exist nineteen non-isomorphism such designs in which λ ∈ {1, 2, 3, 4, 6, 12, 16, 18} and x = psun(q) with (n, q) ∈ {(2, 7),(2, 9),(2, 11),(3, 3),(4, 2)}.