Violation of Bell inequality based on $S_4$ symmetry

In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the violation 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 $S_4$ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of $S_4$ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.