Abstract We provide classification results for translation generalized quadrangles of order less than or equal to 64, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in $$\textrm{PG}(3n-1,2)$$ PG ( 3 n - 1 , 2 ) , for $$n=3,4$$ n = 3 , 4 , and that of the pseudo-ovals in $$\textrm{PG}(3n-1,q)$$ PG ( 3 n - 1 , q ) , for $$n=5,6$$ n = 5 , 6 , such that one of the associated projective planes is Desarguesian.

Abstract The uniqueness of the inversive plane of order sixty-four, up to isomorphism, is established. Equivalently, it is shown that every ovoid of $$\mathrm{PG}(3,64)$$ PG ( 3 , 64 ) is an elliptic quadric.