Point-primitive, line-transitive generalised quadrangles of holomorph type

Let G be a group of collineations of a finite thick generalised quadrangle Gamma. Suppose that G acts primitively on the point set P of Gamma, and transitively on the lines of Gamma. We show that the primitive action of G on P cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Gamma for which the action of G on P is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.