= Solution
Use the <computational basis> ordered as $|000\rangle,|001\rangle,|010\rangle,|011\rangle,|100\rangle,|101\rangle,|110\rangle,|111\rangle$. Since $Z|0\rangle=|0\rangle$ and $Z|1\rangle=-|1\rangle$,
$$
\boxed{S\big|_{N=3}=\operatorname{diag}(3,1,1,-1,1,-1,-1,-3)}.
$$
For arbitrary $N$, a computational basis vector with $q$ ones has $N-q$ contributions $+1$ and $q$ contributions $-1$, giving eigenvalue $N-2q$. All $q=0,\ldots,N$ occur, so
$$
\boxed{\operatorname{spec}S=\{N,N-2,\ldots,-N\},\qquad\dim\ker(S-(N-2q)I)=\binom Nq}.
$$
The <binomial coefficient> counts which $q$ sites are excited. Thus there are exactly $N+1$ distinct <magnetization sectors of a spin chain>, and their dimensions sum to $\sum_q\binom Nq=2^N$, the full Hilbert-space dimension.
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