= Character table of the Klein four-group
{title2=$\chi_{pq}(a^ub^v)=(-1)^{pu+qv}$}
Write the <Klein four-group> as $\{1,a,b,ab\}$. Its four one-dimensional <irreducible representations> independently assign signs to $a,b$, so the displayed <characters> have $p,q\in\{0,1\}$. The <character table> is
$$
\begin{array}{c|rrrr}
&1&a&b&ab\\\hline
\chi_{00}&1&1&1&1\\
\chi_{01}&1&1&-1&-1\\
\chi_{10}&1&-1&1&-1\\
\chi_{11}&1&-1&-1&1
\end{array}.
$$
Every <conjugacy class> is a singleton because the group is abelian. The four rows exhaust the complex irreducibles by the equality between their number and the number of classes. If $H$ is this table, $HH^T=4I$ proves <character orthogonality>. The diagonal three-dimensional representation by half-turns about the coordinate axes is the <direct sum> of the three nontrivial characters, with total character $(3,-1,-1,-1)$.
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