= Solution
Identify the diagonal <Cartan subalgebra> with $\mathbb C^n$ through its coordinates $(h_1,\ldots,h_n)$, and identify real roots with coordinate vectors using the invariant Euclidean <inner product>. A <Weyl reflection> is $s_\alpha(h)=h-2(h,\alpha)\alpha/(\alpha,\alpha)$. It does not change when $\alpha$ is replaced by $-\alpha$. For the <Cn root system>, its explicit actions are
$$
\boxed{\begin{array}{c|c}
\text{root, up to sign}&\text{changed coordinates}\\\hline
2\varepsilon_i&h_i\mapsto-h_i\\
\varepsilon_i-\varepsilon_j&(h_i,h_j)\mapsto(h_j,h_i)\\
\varepsilon_i+\varepsilon_j&(h_i,h_j)\mapsto(-h_j,-h_i)
\end{array}}
$$
Every coordinate not displayed stays fixed. Thus each generating <Weyl reflection> is a signed permutation, so the <Weyl group> is contained in the <signed symmetric group>. Conversely, the first row supplies each independent sign change and the second supplies every coordinate <transposition>; these generate the entire <signed symmetric group>. This proves the <Cn Weyl group> identification, including its size:
$$
\boxed{W\cong(\mathbb Z/2\mathbb Z)^n\rtimes S_n,\qquad |W|=2^nn!.}
$$
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