This is a question we've been given as a practise exam question. We are given the following character table (which is GL_{2}(3), but we can't use the fact that we know which group this is in the question): 
We have the following questions:
1) Find $|G|$ and $|C_{G}(x)|$ for $x=a,b,c,d,e,f,g$.
2) Find all the normal subgroups of $G$.
3) Find $o(x)$ for $x=a,b,c,d,e,f,g$.
For 1), it is easy to see that $|G| = 48$. We can also calculate the size of the centralisers easily. We get them to be $48,4,6,8,6,8,8$ for $a,b,c,d,e,f,g$ respectively
for 2, im aware that a normal subgroup $N = \text{ker}(\chi) = \{ g \in G : \chi (g) = \chi(1) \}$. Using this, i believe we have the following normal subgroups:
$N_{1} = \{1,a,c,d,e\}, N_{2} = \{1,a,d\}, N_{3} = \{1, a \}.$
For 3), I'm quite confused. Firstly, by Cauchy's theorem, we most definitely have an element of order 2 and element of order 3 in the group. Furthermore, since $o(x) | |C_{G}(x)|$, we know $b,d,f,g$ will be powers of $2$. Since we must have an element of order $3$ in the group, this could be either $a,c$ or $e$. From here, I don't know how to use this information to get what I need. Any help would be appreciated.
This is only a partial answer, but perhaps it is helpful.
The fact that there is a unique element in the conjugacy class of $a$ says that every element commutes with $a$, i.e., $a$ is central. And for the same reason, no other element is central. So we have the centre of the group consists of 1 and $a$. This obviously means $a$ is order 2.
The subgroup consisting of 1, the element $a$, and the elements of $d$, is of size 8, and all its elements other than 1 and $a$ are conjugate in $G$. In particular, they all have the same order, and that order divides 8. By looking at the classification of groups of order 8, we see this is either the quaternion group or an elementary abelian 2-group.
But $\chi_4$ defines a 2-dimensional of this group, and we can see that it is irreducible, so the group must not be abelian. Thus it is the quaternion group, and the elements of $d$ have order 4.