= Solution
Take $H=\mathbb C^{S_3}$, the <function Hopf algebra of a finite group>. Multiplication is pointwise, and
$$
(\Delta f)(g,h)=f(gh),\qquad\varepsilon(f)=f(1),\qquad(Sf)(g)=f(g^{-1}).
$$
The group associativity, identity and inverse laws respectively verify coassociativity, the counit and the <antipode>; pointwise products show that $\Delta,\varepsilon$ are algebra homomorphisms. Thus $H$ is a <commutative Hopf algebra>.
For $g=(12)$ and $h=(23)$, using right-to-left permutation composition, $gh=(123)$ whereas $hg=(132)$. The delta function $f=\delta_{(123)}$ has $(\Delta f)(g,h)=1$ but $(\Delta^{\rm op}f)(g,h)=0$. Therefore \b[$\boxed{\mathbb C^{S_3}\text{ is commutative but noncocommutative}}$.]
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