The photoproduction of $omega$ mesons off the proton has been studied in the reaction $gamma pto p,omega$ using the CEBAF Large Acceptance Spectrometer (CLAS) and the frozen-spin target (FROST) in Hall B at the Thomas Jefferson National Accelerator Facility. For the first time, the target asymmetry, $T$, has been measured in photoproduction from the decay $omegatopi^+pi^-pi^0$, using a transversely-polarized target with energies ranging from just above the reaction threshold up to 2.8 GeV. Significant non-zero values are observed for these asymmetries, reaching about 30-40% in the third-resonance region. New measurements for the photon-beam asymmetry, $Sigma$, are also presented, which agree well with previous CLAS results and extend the world database up to 2.1 GeV. These data and additional $omega$-photoproduction observables from CLAS were included in a partial-wave analysis within the Bonn-Gatchina framework. Significant contributions from $s$-channel resonance production were found in addition to $t$-channel exchange processes.
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