Possible Automorphism Groups for an $S(3,5,26)$

Abstract

Let $G$ be the automorphism group of an $(3,5,26)$ design. We show the following: (i) If $13$ divides $|G|$, then $G$ is a subgroup of $Z_2\times F_{r_{13\cdot 12}}$, where $F_{r_{13\cdot 12}}$ is the Frobenius group of order $13\cdot 12$; (ii)If $5$ divides $|G|$, then $G\cong Z_5$ or $G\cong D_{10}$; and (iii) Otherwise, either $|G|$ divides $3\cdot 2^3$ or $2^4$.