The character table of a finite group

Let $G$ be a finite group, let ${\chi }_1,\dots ,{\chi }_r$ be the irreducible complex characters of $G$, one per equivalence class (An irreducible complex character), and let $g_1,\dots ,g_s$ be representatives of the distinct conjugacy classes of $G$ (The conjugacy class ${\mathrm{Cl}}_G(x)$ and centralizer $C_G(x)$ of an element). The character table of $G$ is the array whose rows are indexed by ${\chi }_1,\dots ,{\chi }_r$, whose columns are indexed by $g_1,\dots ,g_s$, and whose $(i,j)$ entry is ${\chi }_i(g_j)$.

The array is written with a top row recording the conjugacy-class sizes $\mathrm{\mid }{\mathrm{Cl}}_G(g_j)\mathrm{\mid }$ above the column labels, and with the first column recording the degrees ${\chi }_i(1)$ beside the row labels. The ordering of the rows and of the columns is arbitrary: permuting either does not change the table as data. Because each ${\chi }_i$ is constant on each conjugacy class, the entry ${\chi }_i(g_j)$ is independent of the choice of representative $g_j$. That the number of rows equals the number of columns is a theorem about the table, proved later on this page as The character table is square and invertible, not part of the present definition.