the sphere. In both cases we know the (quantum) group acting maximally on the sphere. In the first case, this is the orthogonal group $O_n\subset M_n({ {\mathbb R} })$ whereas in the second case [@BanGos], it is Wang’s [@WangOrth] free orthogonal quantum group $O_n^+$. We define an ${\varepsilon}$-version $O_n^{\varepsilon}$ of these objects by $$C(O_n^{\varepsilon})=C^*(u_{ij}, i,j=1,\ldots,n\;|\; u_{ij}=u_{ij}^*, u \textnormal{ is orthogonal}, R^{{\varepsilon}}\textnormal{ hold}),$$ where $R^{\varepsilon}$ are the relations: $$u_{ik}u_{jl}=\begin{cases}
u_{jl}u_{ik}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=1$}\\
u_{jk}u_{il}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=0$}\\
u_{il}u_{jk}&\text{if  ${\varepsilon}_{ij}=0$ and ${\varepsilon}_{kl}=1$}
\end{cases}$$ Moreover, we define an ${\varepsilon}$-version $S_n^{\varepsilon}\subset O_n^{\varepsilon}$ of the symmetric group by $$C(S_n^{\varepsilon})= C^*(u_{ij}\;|\; u_{ij}=u_{ij}^*=u_{ij}^2,\sum_ku_{ik}=\sum_ku_{kj}=1\;\forall i,j, \mathring R^{\varepsilon}\textnormal{ hold}),$$ where $\mathring R^{\varepsilon}$ are the relations: $$u_{ik}u_{jl}=\begin{cases}
u_{jl}u_{ik}&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=1$}\\
0&\text{if  ${\varepsilon}_{ij}=1$ and ${\varepsilon}_{kl}=0$}\\
0&\text{if  ${\varepsilon}_{ij}=0$ and ${\varepsilon}_{kl}=1$}
\end{cases}$$ Note that the quotient of $C(S_n^+)$ by $R^{\varepsilon}$ coincides with the one by $\mathring R^{\varepsilon}$ (Lemma \[LemREpsAeqRRing\]). Again, the choice ${\varepsilon}={\varepsilon}_{\operatorname{comm}}$ yields $O_n^{\varepsilon}=O_n$ and $S_n^{\varepsilon}=S_n$, whereas ${\varepsilon}={\varepsilon}_{\operatorname{free}}$ yields $O_n^{\varepsilon}=O_n^+$ and $S_n^{\varepsilon}=S_n^+$. The quantum groups $S_n^{\varepsilon}$ coincide with Bichon’s quantum automorphism groups of graphs [@BicQAG] and they are quantum subgroups of Banica’s quantum automorphisms of graphs [@BanQAG]. The latter one are given by quotients of $C(S_n^+)$ by the relations $u{\varepsilon}={\varepsilon}u$.

On the level of groups (or monoids) such mixed commutation relations 