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Classifying transitive subgroups of $S_n$ is a classical problem with contributions from many authors. The paper [10.1016/j.jsc.2004.08.002, MR:2168238] classifies these groups for $16\leq n\leq 30$, and confirms results and contains references for $n\leq 15$.

Most data on transitive groups is extracted from or computed with GAP or Magma by J. Jones. Data on integral representations of selected groups was provided by A. Bartel.