Following Baxter's method of producing Q 72 -operator, we construct the Q -operator of the root-of-unity eight-vertex model for the crossing parameter with odd N where Q 72 does not exist. We use this new Q -operator to study the functional relations in 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 (solid-on-solid) 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 the SOS model.