 Character table of the nonabelian group of order twenty-one

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 $\alpha ,\beta =(-1\pm i\sqrt{7})/2$. With suitable class labels the table isOne 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.