= Character table of the symmetry group of a square
{title2=$1^2+1^2+1^2+1^2+2^2=8$}
In the order of the <conjugacy classes of the symmetry group of a square> displayed there, its irreducible complex <character table> is
$$
\begin{array}{c|rrrrr}
&e&r^2&\{r,r^3\}&\{s,r^2s\}&\{rs,r^3s\}\\\hline
A_1&1&1&1&1&1\\A_2&1&1&1&-1&-1\\B_1&1&1&-1&1&-1\\B_2&1&1&-1&-1&1\\E&2&-2&0&0&0
\end{array}
$$
The four <one-dimensional representations> assign independent signs to $r$ and $s$. The remaining row is the <character> of the ordinary planar <representation>: a rotation has trace $2\cos\theta$ and a reflection has trace zero. Weighted by class sizes, these rows satisfy <character orthogonality>. Their squared dimensions sum to eight, so they exhaust the <irreducible representations>.
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