= Character table of the nonabelian group of order twenty-one
{title2=$G=C_7\rtimes C_3$}
For the nontrivial action of $C_3$ on $C_7$, the <conjugacy class> sizes are $1,7,7,3,3$. Three <linear characters> factor through the quotient $C_3$, and the remaining two <irreducible characters> have degree three. Put $\omega=e^{2\pi i/3}$ and let $\alpha,\beta=(-1\pm i\sqrt7)/2$. With suitable class labels the table is
$$
\begin{array}{c|ccccc}
&1&g_2&g_3&g_4&g_5\\\hline
\text{class size}&1&7&7&3&3\\
\chi_1&1&1&1&1&1\\
\chi_2&1&\omega&\omega^2&1&1\\
\chi_3&1&\omega^2&\omega&1&1\\
\chi_4&3&0&0&\alpha&\beta\\
\chi_5&3&0&0&\beta&\alpha
\end{array}
$$
One can derive the last two rows by inducing nontrivial <characters> of $C_7$ along the two orbits of the order-three <automorphism> on their exponents. Their degree-three orbit sums vanish off $C_7$. On a generator they give the two sums of seventh roots over quadratic residues and nonresidues. Their sum is $-1$ and their product is $2$: the nine pairwise exponent sums between the two triples contain zero three times and every nonzero residue once. Thus they are the roots of $X^2+X+2$. Alternatively <column orthogonality> forces the zeros on the <centralizers> of order three, and orthogonality with the identity column interchanges $\alpha$ and $\beta$ in the last row. This finite table illustrates how <centralizer> sizes and a small number of given values determine missing <character> values.
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