A translation plane of order \(7^ 2\) with a very small translation complement
The authors construct a translation plane of order 49\(=7^ 2\) whose translation complement \({\mathcal C}\), the group of collineations fixing the origin (0,0), has order 72 and its factor group modulo the kernel \(K\) is the dihedral group of order 12. The construction is accomplished by exhibiting a set \(S\) of forty-nine 2 by 2 matrices over \(GF(7)\) forming a matrix spread. All calculations are performed on the set \(S\). The authors claim that the plane obtained has the smallest translation complement among all known planes of order 49. No proof is given of this statement, and it may well have been true in 1986 when the article was originally submitted. Since then however several additional translation planes of order 49 have been found.
scientific article; zbMATH DE number 436562