Some finite translation planes arising from \(A_ 6\)-invariant ovoids of the Klein quadric (Q917901)

The authors enumerate (up to isomorphism) all translation planes of order \(p^ 2\), where p is a prime satisfying \(5\leq p\leq 19\), whose translation complement contains a subgroup \(\Gamma\) such that \(\Gamma\) modulo its scalars is isomorphic to \(A_ 6\). The basic technique used in the paper is the Klein correspondence, thereby reducing the problem to a study of \(A_ 6\)-invariant ovoids on the Klein quadric of PG(5,q). A computer was used for the cases \(q=17\) and \(q=19\).