a few duplicates in the first 3 lines of this list; see the previous section for details. ST is the Shephard–Todd number of the reflection group. Rank is the dimension of the complex vector space the group acts on. Structure describes the structure of the group. The symbol * stands for a central product of two groups. For rank 2, the quotient by the (cyclic) center is the group of rotations of a tetrahedron, octahedron, or icosahedron (T = Alt(4), O = Sym(4), I = Alt(5), of orders 12, 24, 60), as stated in the table. For the notation 21+4, see extra special group. Order is the number of elements of the group. Reflections describes the number of reflections: 26412 means that there are 6 reflections of order 2 and 12 of order 4. Degrees gives the degrees of the fundamental invariants of the ring of polynomial invariants. For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6. For more information, including diagrams, presentations, and codegrees of complex reflection groups, see the tables in . Degrees Shephard and Todd proved that a finite group acting on a complex vector space is a complex reflection group if and only if its ring of invariants is a polynomial ring (Chevalley–Shephard–Todd theorem). For being the rank of the reflection group, the degrees of the generators of the ring of invariants are called degrees of W and are listed in the column above headed "degrees". They also showed that many other invariants of the group are determined by the degrees as follows: The center of an irreducible reflection group is cyclic of order equal to the greatest common divisor of the degrees. The order of a complex reflection group is the product of its degrees. The number of reflections is the sum of the degrees minus the rank. An irreducible complex reflection group comes from a real reflection group if and only if it has an invariant of degree 2. The degrees di satisfy the formula Codegrees For being the rank of the reflection group, the codegrees of W can be defined by For a real reflection group, the codegrees are the degrees minus 2. The number of reflection hyperplanes is the sum of the codegrees plus the rank. Well-generated complex reflection groups By definition, every complex reflection group is generated by its reflections.