Translation Planes of Order q6 Admitting SL(2, q2)

In this paper we will give a simple construction for at least q(q + 1)/6e nondesarguesian translation planes GZ of order q6 whenever q =pe with p a prime. The elations of CY fixing the origin 0 generate a group S = SL(2, q*) having q + 2 orbits on the line L, at infinity. The group (Aut a),, has just two orbits on L,, of lengths q* + 1 and q6 q*. The kernel of LPI is GF(q3), and S acts irreducibly on the underlying four-dimensional GF(q3)-space exactly as it does in the case of the desarguesian plane of order q6. The construction was motivated by Example 8.2 of [2]. However, the plane of order 26 constructed there is not the same as the one obtained here. Variations on the construction are undoubtedly possible. The planes also differ from those in [ 11: the kernel and action on L, are quite different for those planes.