A translation plane of order complement 25 with small translation complement.

Abstract

Various translation piant!s of order 25 have so far heen constructed and their translation complements determined. All these planes except the two flag transitive planes of order 25 of Foulser and one  of the exceptional planes of Walker are such that at least one of their matrix representation sets admit non-trivial nuclei. In this paper we construt a new-translation plane of order 25, all of whose matrix representation sets admit only trivial nucleus. Further the translation complement modulo the subgroup of scalar coliineations is a dihedral group of order 24 and is the smallest when compared with all the planes reported so far. The translation complement of this plane divides the set of ideal points into 4 orbits of lengths 4, 4, 6 and 12.