For Odd N

Following Baxter's method of producing Q 72-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the parameter η = 2mK N with odd N where Q 72 does not exist. We use this new Q-operator to study the functional relations through the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure of Baxter's eight-vertex SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q-operator and fusion weights of SOS model.