[Enantiomer]

I'm trying to understand teh difference between irreducible and reducible representations but I'm having a bit of trougle and the book I'm using doesn't really epand enough on it.
two specific examples I'm trying to understand are the reducible representations of water and an O_h group symmetry model.
I know that the reducible representation of water should look like:
E(9) C₂ (-1) sigma_v(xz)(1) and sigma_v(yz)(3)...
but where are these numbers coming from?  how is there 9 forms of E in the water molecule?  I would assume there is only 4 for A1 A2 B1 and B2 but that doesn't seem to be the case?  How do I figure out how to get the numbers for the characters of the reducible representations?
  And the irreducible representations of this C_2v group can be explained through the equations N = 1/h Σ X^x_r * X^x_i * n^xWhere X^x_r is "the character for a particular class of operation" (or simple the character for the E, C₂... etc. in our reducible representation), but here's where I start to get confused, what exactly is X^x_i and n_x?  I'm aware they change for each irreducible species but I don't understand how or why?

Quick edit*:  Okay I think I understand that X^x_i is acctually a part of the character table but I'm still confused as to how I would get that without a character table... or will I always need one/need to make one when doing irreducible representation from a reducible one?
If someone could clear this up to me or point out an article that could help me out I would extremely appreciate it.

[CopperSmurf]

you'll always need the character table. Your reducible E should definitely not be 9, it should be 2.

Reducible representation is based on your point group assigned to the molecule and each symmetry operation you do on it (i.e. mirror plane (Sigma) or rotation).

Your reducible representation is a linear combination of your irreducible representation, so you use that formula as the next step to find out which letters (like A1) belongs to it.

btw, this is usually really poorly taught at schools for some reason, they always assume you know all the quantum mechanics and math behind it all.. which obviously at least 50% of us don't know.

[Enantiomer]

I've been going over it and I'm starting to understand but at the same time I'm still having quite a few problems,
I'm trying to do O_h group symmetry for a cube and I'm having slight problems.  Does anyone know what's wrong with my reducible representation in this scenario?
O_h|E  8C₃  6C₂  6C₄  3C₄²  i  6S₄  8S⁶  3σ_h  6σ_d   (where h = 48)
Γ  |8   0      0      2      2   0   0      0    4     2

I'm thinking that the number of  3σh bonds that stay in place for my reducible representation is wrong but I don't know why, anyone have an idea?