A hemisystem of a nonclassical generalised quadrangle.

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points H such that every line ℓ meets H in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q,q2) were those of the elliptic quadric Q−(5,q), q odd. We show in this paper that there exists a hemisystem of the Fisher-Thas-Walker-Kantor generalised quadrangle of order (5,52), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3⋅A7-hemisystem of Q−(5,5), first constructed by Cossidente and Penttila.