Here is an elementary proof of a related general fact. Let $H$ be any finite nilpotent group with $H^{\prime} = Z(H)$ cyclic of order $m$. Let $z$ be a generator of $Z(H)$. Then $\langle z \rangle $ has a (faithful) linear character $\lambda$ such that $\lambda(z)$ is a complex primitive $m$-th root of unity.

Note then that all irreducible constituents of ${\rm Ind}_{Z(H)}^{H}(\lambda)$ are faithful. For let $\chi$ be one such. Then by Frobenius recipirocity (and Clifford' Theorem), ${\rm Res}^{H}_{Z(H)}(\chi) = \chi(1)\lambda$, so that ${\rm Res}^{H}_{Z(H)}(\chi)$ is certainly faithful. On the other hand, if $\chi$ were not faithful, then ${\rm ker} \chi$ contains a minimal normal subgroup $M$ of $H$. Since $H$ is nilpotent, $M \leq Z(H)$ ( for $M \cap Z(H) \neq 1$ and $M$ is minimal), contrary to the fact that $M \leq {\rm ker} \chi$ and ${\rm Res}^{H}_{Z(H)}(\chi)$ is faithful.

Furthermore, ( as is well-known, and may be found in the character theory text of I.M. Isaacs for example), since $H$ is nilpotent, it follows that if $\theta$ is any faithful irreducible character of $H$, then $\theta$ vanishes identically outside $Z()$. For choose $a \in H \backslash Z(H)$ and choose $b \in H \backslash C_{H}(a)$ we have $[a,b] = w$ for some $1 \neq w \in Z(H)$. Then $b^{-1}ab = wa$. Hence $\theta(a) = \theta(b^{-1}ab) = \theta(wa)$. But by Schur's lemma, $w$ is represented by a scalar matrix in any representation affording character $\theta$, and the scalar, $\alpha$ say, is not $1$ as $\theta$ is faithful . Hence $\theta(a) = \theta(wa) = \alpha \theta(a) $, so that $\theta(a) = 0$.

Hence for our previous faithful irreducible character $\chi$ of $H$, the orthogonality relations yield $|H| = \sum_{ h \in H}|\chi(h)|^{2} = |Z(H)|\chi(1)^{2}$, so that $\chi(1) = \sqrt{[H:Z(H)]}.$

Here is an elementary proof of a related general fact. Let $H$ be any finite nilpotent group with $H^{\prime} = Z(H)$ cyclic of order $m$. Let $z$ be a generator of $Z(H)$. Then $\langle z \rangle $ has a (faithful) linear character $\lambda$ such that $\lambda(z)$ is a complex primitive $m$-th root of unity.

Note then that all irreducible constituents of ${\rm Ind}_{Z(H)}^{H}(\lambda)$ are faithful. For let $\chi$ be one such. Then by Frobenius recipirocity (and Clifford' Theorem), ${\rm Res}^{H}_{Z(H)}(\chi) = \chi(1)\lambda$, so that ${\rm Res}^{H}_{Z(H)}(\chi)$ is certainly faithful. On the other hand, if $\chi$ were not faithful, then ${\rm ker} \chi$ contains a minimal normal subgroup $M$ of $H$. Since $H$ is nilpotent, $M \leq Z(H)$ ( for $M \cap Z(H) \neq 1$ and $M$ is minimal), contrary to the fact that $M \leq {\rm ker} \chi$ and ${\rm Res}^{H}_{Z(H)}(\chi)$ is faithful.

Furthermore, ( as is well-known, and may be found in the character theory text of I.M. Isaacs for example), since $H$ is nilpotent, it follows that if $\theta$ is any faithful irreducible character of $H$, then $\theta$ vanishes identically outside $Z(H)$. For choose $a \in H \backslash Z(H)$ and choose $b \in H \backslash C_{H}(a)$ we have $[a,b] = w$ for some $1 \neq w \in Z(H)$. Then $b^{-1}ab = wa$. Hence $\theta(a) = \theta(b^{-1}ab) = \theta(wa)$. But by Schur's lemma, $w$ is represented by a scalar matrix in any representation affording character $\theta$, and the scalar, $\alpha$ say, is not $1$ as $\theta$ is faithful . Hence $\theta(a) = \theta(wa) = \alpha \theta(a) $, so that $\theta(a) = 0$.

Hence for our previous faithful irreducible character $\chi$ of $H$, the orthogonality relations yield $|H| = \sum_{ h \in H}|\chi(h)|^{2} = |Z(H)|\chi(1)^{2}$, so that $\chi(1) = \sqrt{[H:Z(H)]}.$