I have been reading about induced characters, and I haven't found explicitly written anywhere, the relation between degree of the induced character of a character and the dimension of the character itself. All I know about about induced characters besides the definition) is the Frobenius Character Formula: if $G$ is a group, $H$ a subgroup of $G$ an $\chi_V$ the character of a representation $\rho: \mathbb C \longrightarrow GL(V)$ (for some vector space $V$ over a field $F$), then the induced character $\chi$ of $\chi_V$ is given by the formula $$\chi(g) = \sum_{\sigma \in G/H: x_\sigma g x_\sigma^{-1} \in H} \chi_V(x_\sigma g x_\sigma^{-1})\hspace{5mm} \forall g \in G$$ where in the sum on the right we choose a representative $x_\sigma$ from each right coset $\sigma$ in $G/H$. It seems to me from here that the dimension of the induced character $\chi$ should be $$\chi(1) = \sum_{\sigma \in G/H : x_\sigma g x_\sigma^{-1} \in H} \chi_V(1) = \chi_V(1)|\text{Stab}(H)| = \frac{\dim(V)|G|}{|C(H)|}$$ where $\text{Stab}(H)$ and $C(H)$ denote the stabilizer and set of conjugacy classes of the subgroup $H$ under the conjugation action of $G$ on the set of right cosets of $H$. Is this result correct?
In particular from the last set of equations we see that the induced representation of a linear (degree-$1$) representation $\chi_V$ need not itself be linear, which makes me suspicious of what I wrote in the last paragraph. I would be really obliged if someone could verify if my derivation is correct or explain where I'm going wrong.