Character table from a cyclic quotient and two elementary-abelian orbits ($C_3^2⋊C_4$)

A group of order thirty-six with a normal $C_3^2$ and a cyclic quotient of order four acting freely on its eight nonzero elements has six conjugacy classes of sizes $1,4,4,9,9,9$. Its four linear characters are lifted from the cyclic quotient. The remaining irreducible degrees are four and four, by the sum of squared degrees. Column orthogonality forces the last three values of both degree-four rows to vanish. The square of the order-four action is inversion on $C_3^2$, making the first two nontrivial class values real. Orthogonality then gives sum minus one and product minus two, so the rows are $(4,1,-2,0,0,0)$ and $(4,-2,1,0,0,0)$. This illustrates why reality must be justified when reconstructing a character table from orthogonality.