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In this paper we analyze possible actions of an automorphism of order six ona $2$-$(70, 24, 8)$ design, and give a complete classification for the actionof the cyclic automorphism group of order six $G= \langle \rho \rangle \congZ_6 \cong Z_2 \times Z_3$ where $\rho^3$ fixes exactly $14$ points (blocks) and$\rho^2$ fixes $4$ points (blocks). Up to isomorphism, there are $3718$ suchdesigns. This result significantly increases the number of known$2$-$(70,24,8)$ designs.

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