Classification of rational dihedral groups

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Statement

The following are equivalent for a natural number n:

  1. The Dihedral group (?) of order 2n (and degree n) is a Rational group (?): all its characters over the complex numbers are rational-valued.
  2. The dihedral group of order 2n and degree n is a Rational-representation group (?): all its representations over the complex numbers can be realized over the rational numbers.
  3. \varphi(n) \le 2, where \varphi denotes the Euler phi-function, i.e., the order of the multiplicative group of integers modulo n.
  4. n \in \{ 1,2,3,4,6\}.

Thus, the five dihedral groups that are rational are:

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