On basis invariants of unitary group $W({{J}_{3}}(4))$

In this paper, some properties of basis invariants of the unitary group $W({{J}_{3}}(4))$ of order $336$ generated by reflections in $3$-dimensional unitary space are studied. There is developed a new method of finding in explicit form the basic invariants of group $W({{J}_{3}}(4)).$ This method is based on the following property of group $W({{J}_{3}}(4))$ – group $W({{J}_{3}}(4))$ contains group ${{B}_{3}}$ of symmetries of the cube, and Pogorelov polynomials of the form ${{J}_{{{m}_{i}}}}(G)=\sum\limits_{\sigma \in G}{{{(\vec{x},\sigma\ \vec{s})}^{{{m}_{i}}}}},$ where $G$ is a reflection group, $\sigma$ is reflection with respect to planes of symmetry, $\vec{s}$ is the unit normal vector (with origin $O$) of one of them, vector $\vec{x}$ is given by $\vec{x}=({{x}_{i}}),$ ${{m}_{i}}$ are degrees of the basic invariants of group $G$. In the present paper, using that method, the basis invariants of group $W({{J}_{3}}(4))$ in explicit form were constructed.