Classification of rational dihedral groups
The following are equivalent for a natural number :
- The Dihedral group (?) of order (and degree ) is a Rational group (?): all its characters over the complex numbers are rational-valued.
- The dihedral group of order and degree is a Rational-representation group (?): all its representations over the complex numbers can be realized over the rational numbers.
- , where denotes the Euler phi-function, i.e., the order of the multiplicative group of integers modulo .
Thus, the five dihedral groups that are rational are:
- : cyclic group:Z2 (a degenerate case)
- : Klein four-group (also a degenerate case)
- : Isomorphic to symmetric group:S3.
- : dihedral group:D8
- : , isomorphic to direct product of S3 and Z2