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

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 \(2^4\).