In order to construct a character table (or to solve problems that directly ask for irreducible representations) I have to be able to decompose a representation into irreducible representations. However I don't know how to do this in general.
I understand the definition of an irreducible representation and know it amounts to converting the atrices of the representation into blocks.
The Attempt at a Solution
I have one example available (D3) which is solved by finding transforming the matrices using a transformation matrix consisting of an invariant vector (for example (1,1,1)) and it's orthogonal complement. I doubt this works in general though, or if it does it can get very tedious, but I may be wrong. So, is this the general method, if not, what is (an example to demonstrate would help)?