Can anyone help me to finish the character table following the steps below. (I know how to do part (b) but i'm having trouble with the rest).
Let $G$ be a finite group with conjugacy class representatives $g1, . . . , g6$ having centralizer orders $168, 8, 3, 4, 7, 7$ respectively. Below is a part of the character table of $G$. Find the missing entries of the character table of $G$ by following the steps indicated below.
where $ζ := e^{2πi/7}, A := ζ + ζ^2 + ζ^4$ and $\bar{A}$ is the complex conjugate of $A$.
(a) Find the character of the tensor product of $χ_2$ and its dual. Then decompose it into irreducibles.
(b) Show that $g_4$ has order $4$.
(c) Let $V_2$ be an irreducible $\mathbb{C}G$-module with character $χ_2$. By considering a basis $B$ of $V_2$, such that $[g_5]_{B_i}$ is diagonal, show that $g_5^2$ is conjugate to $g_5$.
(d) Find and decompose the symmetric square of $χ_2$.
(e) Find the missing character.
Any help much appreciated!