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

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 automorphism group of order six $G=⟨\rho ⟩\cong Z_6\cong Z_2\times Z_3$ 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 known $2$-$(70,24,8)$ designs.