Nonabelian groups with \((96,20,4)\) difference sets (Q870064)

Summary: We resolve the existence problem of \((96,20,4)\) difference sets in 211 of 231 groups of order 96. If \(G\) is a group of order 96 with normal subgroups of orders 3 and 4 then by first computing 32- and 24-factor images of a hypothetical \((96,20,4)\) difference set in \(G\) we are able to either construct a difference set or show a difference set does not exist. Of the 231 groups of order 96, 90 groups admit \((96,20,4)\) difference sets and 121 do not. The ninety groups with difference sets provide many genuinely nonabelian difference sets. Seven of these groups have exponent 24. These difference sets provide at least 37 nonisomorphic symmetric \((96,20,4)\) designs.