The inclusive production of the η and ω(782) mesons is measured in the π+π−π0 decay mode in hadronic Z decays and compared to model predictions. The analysis is based on 4 million hadronic Z decays recorded by the ALEPH detector between 1991 and 1995. The η production rate for xp=pmeson/pbeam>0.10 is found to be 0.355±0.011stat±0.024sys per event, and the ω production rate for xp>0.05 is measured as 0.585±0.019stat±0.033sys per event. The branching ratio for ω→μ+μ− is investigated. A total of 18.1±5.9 events are observed, from which the muonic branching ratio is measured for the first time to be BR(ω→μ+μ−)=(9.0±2.9stat±1.1sys)×10−5.