Enumeration of 2-(21, 6, 3) designs with automorphisms of order 7 or 5 (Q2715946)

The authors describe all \(2\)-\((21,6,3)\) designs (each pair in three blocks) which admit an automorphism \(\varphi \) of order \(p\), \(p \geq 5\) a prime. General considerations imply that \(p \leq 7\) and that \(\varphi \) fixes no block or point if \(p =7\). It is proved that if \(p=5\), then \(\varphi \) fixes one point, two blocks and every non-fixed point is contained in at most one fixed block. This information (together with a certain rough classification criterion) is then used for a computer based search. It turns out that the order of the automorphism group of a design can be \(5\), \(10\), \(7\), \(14\), \(21\) or \(63\).