# Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism.

Any hints how to even start?

• I suppose what you are looking for is proposition 25 section 8.2 in serre's representation theory finite groups. – random123 Feb 14 '16 at 18:44
• Look at how an irrep restricts to $C_n$. – Qiaochu Yuan Feb 14 '16 at 20:06
• It follows from the facts that there is abelian subgroup of index $2$ and all irreducible representations of abelian groups have dimension $1$. – Derek Holt May 26 '16 at 10:14

Each irreducible representation of $$D_n$$ is spanned by at most $$[G:H]$$ independent vectors for every abelian subgroup $$H
Let $$H= \subset D_n$$.
$$H$$ is clearly abelian and $$[G:H]=2$$, hence the result.
if $$|H|=n$$ is even, there are 4 irreducible representations of dimension 1 and $$n/2 - 1$$ irreducible representations of dimension 2.