Investigation of solvable (120, 35, 10) difference sets

Abelian difference sets with parameters (120, 35, 10) were ruled out by Turyn in 1965. Turyn's techniques do not apply to nonabelian groups. We attempt to determine the existence of (120,35,10) difference sets in the 44 nonabelian groups of order 120. We prove that if a solvable group admits a (120, 35, 10) difference set, then it admits a quotient group isomorphic to the cyclic group of order 24 or to U24 ≅ (x,y: x⁸ - y³ - 1, xyx ^-1-y^-1}. We describe a computer search, which rules out solutions with a ℤ24 quotient. The existence question remains undecided in the three solvable groups admitting a U₂₄ quotient. The question also remains undecided for the three nonsolvable groups of order 120.