Within Guney-Hillery approach a number of examples of classical and quantum bounds on sum of probabilities resulting from two orbits of
S4 is considered. It is shown that the violation of Bell's inequalities is rather rare and gentle.
The recently proposed (Güney and Hillery in Phys Rev A 90:062121, 2014; Phys Rev A 91:052110, 2015) group theoretical approach to the problem of violating the Bell inequalities is applied to
S4 group. The Bell inequalities based on the choice of three orbits in the representation space corresponding to standard representation of
S4 are derived and their violation is described. The corresponding nonlocal games are analyzed.
In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the breaking of Bell inequalities has been proposed which is based on the theory of finite group representations. We apply here this method to more complicated example of
S4 symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of
S4 determined by the geometry of tetrahedron we find that the Bell inequality under consideration is broken in quantum theory. The corresponding nonlocal game is analyzed.