On symmetric \(2\)-\((70,24,8)\) designs with an automorphism of order \(6\)

Abstract

In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic group of order six \(G= \langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3\), where \(\rho^3\) fixes exactly \(14\) points (blocks) and \(\rho^2\) fixes \(4\) points (blocks). Up to isomorphism there are \(3718\) such designs. This result significantly increases the number of previously known \(2\)-\((70,24,8)\) designs.