Classification of rational dihedral groups

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:

• $n = 1$: cyclic group:Z2 (a degenerate case)
• $n = 2$: Klein four-group (also a degenerate case)
• $n = 3$: Isomorphic to symmetric group:S3.
• $n = 4$: dihedral group:D8
• $n = 6$: $D_{12}$, isomorphic to direct product of S3 and Z2

Related facts

• Classification of rational generalized dihedral groups