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.
Thanks in advance.