Reddit reviews Chemical Applications of Group Theory, 3rd Edition

There's a book titled "Chemical Applications of Group Theory"

https://www.amazon.com/Chemical-Applications-Group-Theory-3rd/dp/0471510947

Cotton's "Chemical Applications to Group Theory" is pretty much the basis for all undergraduate classes that teach group theory. It's expensive though, and probably not the first book you'll want to read on the subject.

I would recommend Bertolucci's "Symmetry and Spectroscopy". It has a lot of great info, and is only $15.

Some good online sources (not all notes are about group theory, so pick and choose what will help you):

http://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/
http://chemistry.caltech.edu/courses/ch112/syllabus.html
Under "Symmetry in Chemistry"

You should also have a working knowledge of matrix algebra. If you want to look into the subject deeper, a good understanding of linear algebra will help.

Cotton's Chemical Applications of Group Theory is a decent resource.

In short, the symmetry of a state is the direct product of the irreducible representations of all of the orbitals occupied in that state. A full orbital only contributes the totally symmetric representation because the direct product of any irreducible representation with itself is the totally symmetric rep. Because of that fact, you only have to really take the direct product of the partially full orbitals to determine the symmetry of a state.

This web page also has some useful information about the octahedral group, including the product tables.

None exist. But if you must, Cotton's book is obviously top notch.
Alternatively, one taught from a math perspective might be good.

https://www.amazon.com/Chemical-Applications-Group-Theory-3rd/dp/0471510947