Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Rosa Maria Miró-Roig[en] Fixed $4 \leq d$ and a primitive $d$th root of unity $e$ we consider the ideal $I_{d}$ generated by all the $\mu$ monomials of degree $d$ invariant under the action of the diagonal matrix $M= Diag(1,e, e^{2},e^{3})$. We prove that $I_{d}$ is a monomial Galois Togliatti system ($GT$-system). We study the variety $F_{d}$ image of the Galois covering $\varphi_{Id}$ : $\mathbb{P}^{3}\rightarrow mathbb{P}^{\mu-1}$ with cyclic Galois group $\mathbb{Z}/d$ associated to $I_{d}$. We call this 3-dimensional variety $GT$-threefold. Finally, we demonstrate that the homogeneous ideal of $GT$-threefolds is a lattice ideal associated to a saturated partial character from $\mathbb{Z^\mu}$.