I am working through a question on an old character theory exam. I've answered the first two parts ok, but am now struggling on the third part. Here is the part that I can't do:
I've computed the character table, and I get
$$\begin{array}{|c|c|c|c|c|c|} \hline & e & g_2 & g_3&g_4&g_5\\ \hline \chi_1 & 1 & 1& 1 & 1& 1\\ \hline \chi_2 & 1 & 1& -1 & -1& 1\\ \hline \chi_3 & 3 & -1& -1 & 1& 0\\ \hline \chi_4 & 3 & -1& 1 & -1& 0\\ \hline \chi_5 & 2 & 2& 0 & 0& -1\\ \hline \end{array}$$
so now I'm just confused about parts (ii) and (iii).
I know that every normal subgroup is an intersection of kernels of some of the irreducible characters, and that $\ker\chi_5=g_2^G$ (the second conjugacy class), but I'm not too sure where exactly to go from here. Any hints would be much appreciated!