  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 $G= \langle \rho \rangle \cong
Z_6 \cong Z_2 \times Z_3$ 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 known
$2$-$(70,24,8)$ designs.

Combinatorics