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Following Baxter's method of producing Q_{72}-operator, we construct theQ-operator of the root-of-unity eight-vertex model for the crossing parameter$\eta = \frac{2m K}{N}$ with odd $N$ where Q_{72} does not exist. We use thisnew Q-operator to study the functional relations in the Fabricius-McCoycomparison between the root-of-unity eight-vertex model and the superintegrableN-state chiral Potts model. By the compatibility of the constructed Q-operatorwith the structure of Baxter's eight-vertex (solid-on-solid) SOS model, weverify the set of functional relations of the root-of-unity eight-vertex modelusing the explicit form of the Q-operator and fusion weights of SOS model.Comment: Latex 28 page; Typos corrected, minor changes in presentation,  References added and updated-Journal versio

TheQ-operator and functional relations of the eight-vertex model at root-of-unity \eta = \frac{2m K}{N} for oddN