this paper is the following: What are all the finite dimensional representations of $V_{4}$ over ${\mathbb{F}_2}$? That is, can one describe all possible actions of the group $V_{4}$ on finite dimensional vector spaces $W$ over ${\mathbb{F}_2}$. Although, we work over an arbitrary field of characteristic two, not much is lost if the reader assumes through out that the ground field $k$ is ${\mathbb{F}_2}$. The reason for restricting to fields of characteristic $2$ is due to the fact that when the characteristic of the ground field $k$ is either zero or odd, the finite dimensional representations $W$ of $V_{4}$ have a very simple nature. Namely, $W$ is a sum of one dimensional representations. One each of these one-dimensional subspaces, the generators $\sigma$ and $\tau$ act as multiplication by $1$ or $-1$. We therefore stick with fields of characteristic $2$. This bring us to the world of modular representations. (That is, the characteristic of the field is a positive divisor of the order of the group.)

Note that $V_4$ is a product of two cyclic groups of order two. In terms of generators and relations, $V_4$ has the following presentation. $$V_4 =   \langle \sigma, \tau \; | \; \sigma^2 = \tau^2 = 