Let L be a thick generalized quadrangle and let G be a group of automorphisms of L. If G acts transitively on the set of non-degenerate ordered pentagons, then L is one of the classical generalized quadrangles W(q), Q(4,q), Q(5,q) or H(3,q(2)). The possibilities for G in each case are determined. We do not use the classification of the finite simple groups (from which this result also follows). [References: 8]
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