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

TL;DR

This paper classifies symmetric 2-(70,24,8) designs with an automorphism of order 6, revealing 3718 such designs and expanding the known catalog of these combinatorial structures.

Contribution

It provides a complete classification of 2-(70,24,8) designs with a cyclic automorphism of order 6, including detailed automorphism action analysis.

Findings

Identified 3718 non-isomorphic designs with the automorphism group of order 6.

Detailed automorphism action analysis on points and blocks.

Significantly increased the known examples of these designs.

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 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.