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  "body": "\n## Definition\n\nLet $G$ be a finite [[group|G5S1M9]].\nLet $\\chi_1, \\ldots, \\chi_r$ be the distinct irreducible complex [[characters|RMWMYBY8TC]] of $G$, and let $C_1, \\ldots, C_s$ be the [[conjugacy classes|0DEBXAJ5SA]] of $G$, with $g_j \\in C_j$.\nThe *character table* of $G$ is the $r \\times s$ matrix with entries $\\chi_i(g_j)$.\nBy convention $\\chi_1$ is the trivial character and $C_1 = \\set{1}$.\n\n## Remarks\n\nThe table is square: $r = s$.\nExample: the symmetric group $S_3$ has three classes, of the identity, the transpositions and the $3$-cycles.\nIts three irreducible characters are the trivial one, the sign, and a character of degree $2$.\nThe last one is the permutation character on three points minus the trivial character.\nIts table is\n$$\\begin{array}{c|ccc} \u0026 1 \u0026 (1\\,2) \u0026 (1\\,2\\,3) \\\\ \\hline \\chi_1 \u0026 1 \u0026 1 \u0026 1 \\\\ \\chi_2 \u0026 1 \u0026 -1 \u0026 1 \\\\ \\chi_3 \u0026 2 \u0026 0 \u0026 -1 \\end{array}$$\nThe 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|AY0B20QC04]].\nNon-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 $\\chi_3$.\nNon-isomorphic groups can have the same character table: the dihedral group of order $8$ and the quaternion group do.\n",
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