= Character table of the quaternion group
The <quaternion group> has <conjugacy classes> $\{1\}$, $\{-1\}$, $\{\pm i\}$, $\{\pm j\}$, $\{\pm k\}$, with sizes $1,1,2,2,2$. Over the complex numbers, the number of classes and the <sum of squares of irreducible representation dimensions> force four one-dimensional representations and one two-dimensional representation. For $\epsilon,\eta\in\{\pm1\}$, the linear character rows on these classes are $(1,1,\epsilon,\eta,\epsilon\eta)$. Orthogonality then gives the remaining row $(2,-2,0,0,0)$. A representation with character $(4,-4,0,0,0)$ contains two copies of the two-dimensional irrep and none of the linear irreps, by the <character inner product>. The choice of scalar field matters: the corresponding quaternionic four-dimensional real representation is irreducible over the reals.
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