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If $A$ is the character table and $A^\ast$ is its conjugate transpose, then the orthogonality relations tell us that $A A^\ast = \text{diag}\{|C_G(g)|\} $, where the enties run over a fixed choice of elements of $G$, one from each conjugacy class. Thus $|\Delta| = \det A A^\ast = \prod |C_G(g)|$ is an integer. On the other hand, $\Delta$ must be rational. This follows from the fact that the action of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ permutes the columns of $A$, hence fixes $\Delta = (\det A)^2$. Thus $\Delta=\pm |\Delta|$ is an integer.