In

- Wang, Shuzhou. Quantum symmetry groups of finite spaces. Comm. Math. Phys. 195 (1998), no. 1, 195--211. MR1637425 (99h:58014), link

a quantum version of the symmetric group $\mathbb{S}_n$ is defined.

Let me sketch Wang's construction.

Let $u_{ij}$ be the characteristic function of the set of $\sigma\in\mathbb{S}_n$ such that $\sigma(j)=i$.

Assume that all entries $u_{ij}$ are projections, and
on each row and column of $u=(u_{ij})$ these projections are orthogonal, and sum up to $1$.
Then the commutative $C^*$-algebra generated by these $u$ is $C(\mathbb{S}_n)$.

Now drop the commutativity condition and let $A_s(n)$ be the $C^*$-algebra generated by all the $u_{ij}$. Then we have a *quantum* analogue of $\mathbb{S}_n$.

It turns out that $A_s(n)$ is a finitely generated Hopf algebra.

The group $\mathbb{S}_n$ acts on an set $X=[1,2,...,n]$ with $|X|=n$. The corresponding action map $(i,\sigma)\mapsto \sigma(i)$ gives by transposition a certain morphism $\alpha$ ($\alpha$ is called *coaction*). This coaction can be expressed as
$\alpha(\delta_i)=\sum\delta_j\otimes u_{ji}$. Furthermore, $\alpha$ is a sort of universal coaction.

It is possible to prove that the following diagram is commutative
$$
\begin{array}{ccc}
C(X) & \to & C(X)\otimes A_s(n)\\\\
\downarrow & & \downarrow\\\\
C(X) & \to & C(X)\otimes C(\mathbb{S}_n)
\end{array}
$$

Furthermore, $C(\mathbb{S}_n)=A_s(n)$ if $n=1,2,3$. For $n\geq4$, $A_s(n)$ is not commutative and infinite dimensional.

For a nice survey about quantum permutation groups and some applications see the following paper:

- Banica, Teodor; Bichon, Julien; Collins, Benoît. Quantum permutation groups: a survey. Noncommutative harmonic analysis with applications to probability, 13--34, Banach Center Publ., 78, Polish Acad. Sci. Inst. Math., Warsaw, 2007. MR2402345 (2009f:46094), link

For a quantum version of the automorphism group of finite graphs (and a quantum version of the dihedral group $\mathbb{D}_4$):

- Bichon, Julien. Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc. 131 (2003), no. 3, 665--673 (electronic). MR1937403 (2003j:16049), link

A complete classification of quantum permutation groups acting on 4 points was given in:

- Banica, Teodor; Bichon, Julien. Quantum groups acting on 4 points. J. Reine Angew. Math. 626 (2009), 75--114. MR2492990 (2010c:46153), link