= Solution
Take the <group algebra> $H=\mathbb C[S_3]$ with
$$
\Delta(g)=g\otimes g,\qquad\varepsilon(g)=1,\qquad S(g)=g^{-1}
$$
on its group basis, extended linearly. Multiplication of group basis elements makes $\Delta,\varepsilon$ algebra homomorphisms, and $g^{-1}g=gg^{-1}=1$ verifies the <antipode> identities. The coproduct is fixed by the tensor flip on every basis element, so this is a <cocommutative Hopf algebra>. But $(12)(23)$ and $(23)(12)$ are distinct basis elements, proving noncommutativity. Thus \b[$\boxed{\mathbb C[S_3]\text{ is noncommutative but cocommutative}}$.]
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