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We characterise the primitive 2-closed groups $G$ of rank at most four thatare not the automorphism group of a graph or digraph and show that if thedegree is at least 2402 then there are just two infinite families or$G\leqslant \mathrm{A}\Gamma\mathrm{L}_1(p^d)$, the 1-dimensional affinesemilinear group. These are the first known examples of non-regular 2-closedgroups that are not the automorphism group of a graph or digraph.

On primitive $2$-closed permutation groups of rank at most four