The 3 4 S + α system has been studied both experimentally and theoretically. Excitation functions of the 3 4 S(α,γ) 3 8 Ar reaction have been measured over the alpha bombarding energy range E α = 3.4-4.4 MeV. The low background provided by the 100% enriched (implanted) target allowed one to observe 10 new resonances, some of them attributed previously to the 3 2 S contamination of the target. Decay schemes, spin-parity values, reduced electromagnetic transition probabilities and resonance strengths have been determined. A semimicroscopic algebraic 3 4 S + α cluster model has been used to describe the new resonances together with the low-lying states of the 3 8 Ar nucleus. This analysis indicates that the J π = 1 - resonances may not be good candidates for the first state of a K π = 0 - band predicted to start in the E x ≃ 10-11 MeV region.