Character table

Remarks

The table is square: $r = s$. Example: the symmetric group $S_3$ has three classes, of the identity, the transpositions and the $3$-cycles. Its three irreducible characters are the trivial one, the sign, and a character of degree $2$. The last one is the permutation character on three points minus the trivial character. Its table is $$\begin{array}{c|ccc} & 1 & (1\,2) & (1\,2\,3) \\ \hline \chi_1 & 1 & 1 & 1 \\ \chi_2 & 1 & -1 & 1 \\ \chi_3 & 2 & 0 & -1 \end{array}$$ The check: $1^2 + 1^2 + 2^2 = 6 = \abs{S_3}$, and the rows are orthogonal for the weighted inner product of the Schur orthogonality relations. Non-example: the $2 \times 3$ matrix with rows $\chi_1, \chi_2$ only is not the character table of $S_3$, because it omits the irreducible character $\chi_3$. Non-isomorphic groups can have the same character table: the dihedral group of order $8$ and the quaternion group do.