In this note, we study Togliatti systems generated by invariants of the dihedral group D2d acting on k[x0, x1, x2]. This leads to the first family of non-monomial Togliatti systems, which we call GT-systems with group D2d. We study their associated varieties SD2d, called GT-surfaces with group D2d. We prove that there are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, I(SD2d), is minimally generated by quadrics and we find a minimal free resolution of I(SD2d).
