Considering a \({\mathbb {Z}}_3\) -graded 3-dimensional space, we develop \({\mathbb {Z}}_3\) -graded differential calculi on the algebra of functions defined on this space, which is a \({\mathbb {Z}}_3\) -graded Hopf algebra. We demonstrate that these differential calculi admit a \(*\) -operation. Additionally, we discover a new \({\mathbb {Z}}_3\) -graded upper triangular quantum group and explicitly define its \({\mathbb {Z}}_3\) -graded Hopf algebraic structure. We also show in detail that this quantum group is the symmetry group of one of these calculi.
