Character of $S_3$ I am trying to learn about the characters of a group but I think I am missing something.
Consider $S_3$. This has three elements which fix one thing, two elements which fix nothing and one element which fixes everything. So its character should be $\chi=(1,1,1,0,0,3)$ since the trace is just equal to the number of fixed elements (using the standard representation of a permutation matrix).
Now I think this is an irreducible representation, so $\langle\chi,\chi\rangle$ should be 1. But it's $\frac{1}{6}(1+1+1+9)=2$. 
So is the permutation matrix representation actually reducible? Or am I misunderstanding something?

The permutation representation is reducible. It has a subrepresentation spanned by the vector $(1,1,1)$. Hence, the permutation representation is the direct sum of the trivial representation and a $(n-1=2)$-dimensional irreducible representation.
