Some translation planes admitting SL(2,5)\(\times SL(2,5)\) (Q1113081)

Translation planes of orders \(31^ 2\) and \(59^ 2\) are constructed with the property that the direct product \(G=SL(2,5)\times SL(2,5)\) is a subgroup of the translation complement. The technique is a generalization of the method used by \textit{H. Lüneberg} [Translation Planes (1980; Zbl 0446.51003)] to construct his \(R^*p\) planes. There are four planes of order \(31^ 2\) and three planes of order \(59^ 2\), and in each plane the two direct factors of G are groups of affine homologies. The spread sets for these planes are given and were obtained using the computer. The author does not show that the planes are new; it is also possible that they are obtainable from known planes by derivation or some other known method. Since few planes of order \(31^ 2\) are known those of the author probably are new, but it would not be surprising if the planes of order \(59^ 2\) are obtainable from known planes.