  In recent papers, the theory of representations of finite groups has been
proposed to analyzing the violation of Bell inequalities. In this paper, we
apply this method to more complicated cases. For two partite system, Alice and
Bob each make one of $d$ possible measurements, each measurement has $n$
outcomes.
  The Bell inequalities based on the choice of two orbits are derived. The
classical bound is only dependent on the number of measurements $d$, but the
quantum bound is dependent both on $n$ and $d$. Even so, when $d$ is large
enough, the quantum bound is only dependent on $d$. The subset of probabilities
for four parties based on the choice of six orbits under group action is
derived and its violation is described. Restricting the six orbits to three
parties by forgetting the last party, and guaranteeing the classical bound
invariant, the Bell inequality based on the choice of four orbits is derived.
Moreover, all the corresponding nonlocal games are analyzed.
