  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
$\eta = \frac{2m K}{N}$ 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 SOS model.
