A translation plane of order 72 with a very small translation complement

A translation plane of order 7² is constructed whose left nucleus is a cyclic group of order 6. Generally the translation complement of a translation plane is quite large when compared to the order of the left nucleus. The plane discussed in this paper has the distinguished feature that its translation complement modulo the scalar collineations group is very small. In fact it is a Dihedral group of order 12 and is the smallest of all the known examples. The translation complement divides the set of ideal points into 9 orbits of lengths 1,1, 6, 6, 6, 6, 6, 6, and 12.