The set of reflections is not a minimal generating set, however, and every irreducible complex reflection groups of rank has a minimal generating set consisting of either or reflections. In the former case, the group is said to be well-generated. The property of being well-generated is equivalent to the condition for all . Thus, for example, one can read off from the classification that the group is well-generated if and only if p = 1 or m. For irreducible well-generated complex reflection groups, the Coxeter number defined above equals the largest degree, . A reducible complex reflection group is said to be well-generated if it is a product of irreducible well-generated complex reflection groups. Every finite real reflection group is well-generated. Shephard groups The well-generated complex reflection groups include a subset called the Shephard groups. These groups are the symmetry groups of regular complex polytopes. In particular, they include the symmetry groups of regular real polyhedra. The Shephard groups may be characterized as the complex reflection groups that admit a "Coxeter-like" presentation with a linear diagram. That is, a Shephard group has associated positive integers and such that there is a generating set satisfying the relations for , if , and where the products on both sides have terms, for . This information is sometimes collected in the Coxeter-type symbol , as seen in the table above. Among groups in the infinite family , the Shephard groups are those in which . There are also 18 exceptional Shephard groups, of which three are real. Cartan matrices An extended Cartan matrix defines the unitary group. Shephard groups of rank n group have n generators. Ordinary Cartan matrices have diagonal elements 2, while unitary reflections do not have this restriction. For example, the rank 1 group of order p (with symbols p[], ) is defined by the matrix . Given: . References Hiller, Howard Geometry of Coxeter groups. Research Notes in Mathematics, 54. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp. * Coxeter, Finite Groups Generated by Unitary Reflections, 1966, 4. The Graphical Notation, Table of n-dimensional groups generated by n Unitary Reflections. pp. 422–423 External links MAGMA Computational Algebra System page Reflection groups Geometry Group theory