Let G be a finite group acting primitively on the sets Ω₁ and Ω₂. We describe a construction of 1-designs with block set Ω₁ and block set Ω₂, having G as an automorphism group. Applying this construction method we obtain a unital 2-(q³+1, q+1, 1), and a semi-symmetric (q⁴−q³+q², q²−q, (1)) from the unitary group U₃(q), where q = 3, 4, 5, 7. From the unital and the semi-symmetric design we build a projective plane PG(2, q²). Further, we describe other combinatorial structures constructed from these unitary groups and structures constructed from U₄(2), U₄(3) and L₂(49).

We also construct self-orthogonal codes obtained from the row span over ₂ or ₃ of the incidence (resp. adjacency) matrices of mostly self-orthogonal designs (resp. strongly regular graphs) defined by the action of the simple unitary groups U₃(q) for q = 3, 4, 7 and U₄(q) for q = 2, 3 and the linear group L₂(49) on the conjugacy classes of some of their maximal subgroups. Some of the codes are optimal or near optimal for the given length and dimension.