# Examples of decomposition representation

Here is a question in the book "Representation theory of finite group, an introductory approach" of Benjamin Steinberg. (Question 3.8(2), page 25) that I need some hints from you :

Give an example of a finite group $G$ and a decomposable representation $\phi : G\to GL_4(\mathbb{C})$ such that the $\phi_g$ with $g\in G$ do not have a common eigenvector.

I tried to give some examples with cyclic groups of order 3, or abelian group of order 4, but I did not succeed. Please give me some examples that you know.

Slightly generalising mike's comment: you certainly don't want your representation to have one-dimensional direct summands. So your only chance is to construct a direct sum of two irreducible two-dimensional representations. Conversely, if a representation is irreducible and more-than-1-dimensional, then the $\phi_g$ cannot have a common eigenvector, since that would generate a subrepresentation. I am sure you will be able to take it from there. – Alex B. Apr 21 '12 at 13:02