The Transitive Groups of Degree Fourteen and Fifteen

All transitive groups of degree fourteen and fifteen up to conjugacy in the symmetric group are listed. The computations, carried out with Cayley, involve the use of subgroup lattices of quotient groups, preimages of the subgroups of the quotient, low index subgroups, systems of imprimitivity, and testing conjugacy of subgroups. Our results for degree fourteen agree with previous results of Miller in 1897, but the results for degree fifteen are new.