(Part of the book posted for reference)
My question is about the paragraph in the bottom of the second image that the author claimed and proved that $\lambda^{N}\in \text{Irr}(N)$, but in my own calculation of its inner product in $\mathbb{C}[G]$( or say $\mathbb{C}[N]$) : $$[\lambda^{N},\lambda^{N}]_N=\frac{1}{|N|}\sum_{x\in N}\lambda^{N}(x)\overline{\lambda^{N}(x)}=\frac{1}{|N|}\sum_{x\in C}\frac{|N|^2}{|C|^2}\lambda(1)=\frac{|N|}{|C|}=2>1$$ where $C=C_{G}(P'), N=N_{G}(P')$ with $P'$ denoting the derived (commutator) subgroup of $P$; $\lambda$ is a linear character of $C$ and $\lambda^N$ denoting the character induced to $N$.