Generalized Quadrangles Associated with G2(q)

In addition to the generalized quadrangles arising naturally from smalldimensional symplectic, unitary, or orthogonal geometries, the only known finite examples are the following quadrangles and their duals: ones with parameters 4, 4 (where 4 = 2e > 8) or q*, 4 (where q = 22e+l > 8) due to Tits [4, p. 3041; and others with parameters q 1, q + 1 (for prime powers q) due to Hall [5] and Ahrens and Szekeres [l]. In this note we will discuss a procedure for constructing all but the last of the above examples, and others as well. In particular, we will prove the following result.