A search for exotic mesons in the π+π+π− system photoproduced by the charge exchange reaction γp→π+π+π−(n) was carried out by the CLAS Collaboration at Jefferson Lab. A tagged-photon beam with energies in the 4.8 to 5.4 GeV range, produced through bremsstrahlung from a 5.744 GeV electron beam, was incident on a liquid-hydrogen target. A partial wave analysis was performed on a sample of 83 000 events, the highest such statistics to date in this reaction at these energies. The main objective of this study was to look for the photoproduction of an exotic JPC=1−+ resonant state in the 1 to 2 GeV mass range. Our partial wave analysis shows production of the a2(1320) and the π2(1670) mesons, but no evidence for the a1(1260), nor the π1(1600) exotic state at the expected levels. An upper limit of 13.5 nb is determined for the exotic π1(1600) cross section, less than 2% of the a2(1320) production.