1,  \sigma \tau = \tau \sigma  \rangle .$$ The group algebra $kV_4$ is then isomorphic to $$k[a , b]/(a^2, b^2),$$ where $a$ corresponds to $1+ \sigma$ and $b$ to $1+\tau$. We define an ideal $U$ of the $kV_4$ as the ideal generated by $a$ and $b$. This is an extremely important ideal called the augmentation ideal of our group ring. Sometime it will be convenient to divide $kV_4$ by ideal generated by $ab$. This simply amounts adding further relation $ab = 0$. The reason for this is that often $ab$ acts on our vector spaces as $0$ and therefore why not simplify our ring even further and add the relation $ab= 0$? We still call the image of $U$ in this new ring $U$ as we do not want to make our notation too complicated. In this paper we present a rather accessible proof of the well-known classification of all the finite dimensional representations of $V_{4}$, or equivalently, of the finitely generated indecomposable $kV_{4}$-modules. These are also known as the modular representations of $V_{4}$. Note that a $V_{4}$ representation where $ab$ acts as zero can be viewed as a finite dimensional $k$-linear space equipped with a pair commuting linear maps 