= Solution
In a one-dimensional <representation>, the relation $srs=r^{-1}$ implies $\rho(r)=\rho(r)^{-1}$, so $\rho(r)=\delta\in\{\pm1\}$. Independently $\rho(s)=\epsilon\in\{\pm1\}$. These give four <irreducible representations>
$$
\chi_{\epsilon,\delta}(r^ks^\ell)=\delta^k\epsilon^\ell.
$$
A two-dimensional <representation> $E$ is obtained by restricting the given <matrices> to the $yz$ plane:
$$
E(r)=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad
E(s)=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
$$
The <eigenvalues> of $E(r)$ are $i$ and $-i$. Its two <eigenlines> are interchanged by $E(s)$, so neither is invariant under the whole <group>. Thus $E$ is a complex <irreducible representation>.
Taking <traces> gives the <character table>, with class sizes included for the <character inner product>:
$$
\boxed{\begin{array}{c|rrrrr}
&C_0&C_1&C_2&C_3&C_4\\
|C_j|&1&1&2&2&2\\\hline
\chi_{+,+}&1&1&1&1&1\\
\chi_{+,-}&1&1&1&-1&-1\\
\chi_{-,+}&1&1&-1&1&-1\\
\chi_{-,-}&1&1&-1&-1&1\\
\chi_E&2&-2&0&0&0
\end{array}.}
$$
For example $\langle\chi_E,\chi_E\rangle=(4+4)/8=1$, and its inner product with each one-dimensional row is $(2-2)/8=0$. The four one-dimensional rows are mutually orthonormal by direct weighted summation. Their squared <dimensions> sum to $4\cdot1^2+2^2=8=|G|$, and there are five rows for five <conjugacy classes>. This proves completeness of the list.
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