= Solution
With $R=C_2(z)$, $S=\sigma_{xz}$ and $T=\sigma_{yz}$, composition gives
$$
\begin{array}{c|cccc}
&E&R&S&T\\\hline
E&E&R&S&T\\
R&R&E&T&S\\
S&S&T&E&R\\
T&T&S&R&E
\end{array}
$$
Every entry belongs to the set, proving closure. $E$ is the <identity element>; every element is its own inverse. Associativity follows because these are actual spatial transformations: composing functions is associative. Thus all group axioms hold, and the group is isomorphic to the <Klein four-group>.
For a concrete matrix check, use $E=\operatorname{diag}(1,1,1)$, $R=\operatorname{diag}(-1,-1,1)$, $S=\operatorname{diag}(1,-1,1)$ and $T=\operatorname{diag}(-1,1,1)$. Their matrix products reproduce the table.
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