  In this note, we study Togliatti systems generated by invariants of the
dihedral group $D_{2d}$ acting on $k[x_{0},x_{1},x_{2}]$. This leads to the
first family of non monomial Togliatti systems, which we call $GT-$systems with
group $D_{2d}$. We study their associated varieties $S_{D_{2d}}$, called
$GT-$surfaces with group $D_{2d}$. We prove that they are arithmetically
Cohen-Macaulay surfaces whose homogeneous ideal, $I(S_{D_{2d}})$, is minimally
generated by quadrics and we find a minimal free resolution of $I(S_{D_{2d}})$.
