Ok so I have learned some very basic things about groups and matrix representations of groups. I have learned that it can be possible to find a "minimal basis" or "irreducible basis" for which a representation corresponds to a linear base change where the representation matrices in the new basis are blocks along the diagonal.
Then from another course I have learned of the Jordan decomposition which is always possible to do for matrices with elements in $\mathbb{C}$. In this decomposition the blocks are always upper triangular.
So for any group with a representation matrix with elements in the complex numbers, we should always be sure to be able to build an irreducible representation where the blocks are triangular?
Does this make sense or am I missing anything? If I am right, is there any special property of the complex numbers which make this possible?