Prove or disprove the following: if we consider the character table of $G$ as a matrix $A$ and $g_1,\cdots, g_m$ is a list of representatives of the conjugacy classes of $G,$ then the matrix A satisfies $\det A = \pm \prod_{i=1}^m |C_G(g_i)|$ (the determinant is only well-defined up to sign as the order of the rows or columns of the character table isn't fixed).
I'm not sure how to prove the statement. I know that the rows of the character table are orthogonal. I think the statement might be false because there likely isn't such a nice formula for computing the determinant of the character table. It might be useful to consider the character table for the regular representation of a group G.