complete understanding of an innocent looking group on four elements would take the works of some of the great minds of the 19th, 20th, and 21st centuries.

Before going further, we remind the reader some basis facts and terminology. We refer the reader to Carlson lecture notes for basic representation theory [@carlson-modulesandgroupalgebras]. In the category of modules over a the Klein group (or more generally, over a $p$-group), the three terms “injective", “projective" and “free" are synonymous. Given a $V_{4}$-module $M$, its Heller shift $\Omega( M)$ is defined to be the kernel of a minimal projective cover of $M$. It can be shown that minimal projective covers are unique up to isomorphism and from that it follows that $\Omega(M)$ is well-defined. Inductively one defines $\Omega^n (M)$ to be $\Omega( \Omega^{n-1} M)$. Similarly, $\Omega^{-1} M$ is defined to be the cokernel of an injective envelope of $M$, and $\Omega^{-n}
(M)$ to be $\Omega^{-1}( \Omega^{-n+1} M)$. Again one can shown that these are well-defined modules. The modules $\Omega^i M$ are also known as the syzygies of $M$. By the classical Krull-Remak-Schmidt theorem, one knows that every representation of a finite group decomposes as a direct sum of indecomposable ones. Thus it suffices 