I'm struggling with a group theory question on the symmetries of the water molecule. First off, I'm giving the following image of the molecule in question:
Now, I've constructed the character table of this molecule as follows, where c denotes the rotations and b denotes the reflections that preserve the symmetry of the molecule. Hence, these symmetries form a group called $G_W = {(c,b | c^2 =b^2 =(bc)^2 = e)} $.
Now I'm asked to provide the three dimensional vector representation $D^V$ of the group $G_W$. The solutions state only the following, without explanation, and I am not sure how they got to this answer.
Can anyone maybe provide some explanation? Thanks in advance!