Let G be a group coacting on an Artin–Schelter regular algebra A homogeneously and inner faithfully. When the identity component A_{e} is also Artin–Schelter regular, providing a generalization of the Shephard–Todd–Chevalley theorem, we say that G is a dual reflection group for A . We give two examples of dual reflection groups of order 16 and study algebraic and geometric properties of three associated Artin–Schelter regular algebras of dimension four.