I have reached the end of a course in representation theory (just an introduction), and I am just doing some examples of finding character tables.
The way I generally do it is to guess a representation and find its character. Assuming I've done well and get an irreducible, I then want to use this and the trivial character to make a new representation (which may or may not be irreducible).
My question is: What can I do with two representations to get new ones?
I know that you can multiply by linear characters. Can I add and subtract to get what I want?
Things you are allowed to do:
Multiply characters (corresponds to tensor product)
Add characters (direct sum)
Take complex conjugate of characters
Then you can use these facts and known representations to subtract and break down characters.
To see if a known irreducible character is a factor of another character, use the inner product and orthogonality laws.