Let $G$ be a finite group with cardinality $m$ and let $IrrG=\{\chi_1,...,\chi_r\}$ be the set of all irreducible characters of $G$ in $\mathbb{C}$. Since $G$ is finite, then $\chi_i(g)=\sum \zeta_j$ with $\zeta_j$ $m$-th root of unity.
If we denote $G$ as $$G=\{g_1,...,g_m\},$$ we can consider the extension of $\mathbb{Q}$ \begin{align*} &K_i=\mathbb{Q}\big(\chi_i(g_1),...,\chi_i(g_m)\big) \\ &K=\mathbb{Q}\big(\chi_1(g_1),...,\chi_1(g_m),\chi_2(g_1),...,\chi_r(g_m)\big) \end{align*} They are subfields of $Split_{\mathbb{Q}}(x^{m}-1)$. So $K_i/\mathbb{Q}$ and $K/\mathbb{Q}$ are Galois extension.
Now we have \begin{align*} &\mathcal{G}_i=\mathcal{G}al(K_i/\mathbb{Q})\\ &\mathcal{G}=\mathcal{G}al(K/\mathbb{Q}) \end{align*} the Galois groups of this extension.
Are there some results which state a connection between $G$ and $\mathcal{G}$?