Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the [AnyonWiki](https://anyonwiki.github.io/pages/Lists/losmffc.html). Most of the fusion rings belonging to these categories belong to one of the following classes: 

* **Finite group rings**, e.g., cyclic groups of order $1,\ldots,7$, $D_3$, etc
* **Representation rings of finite groups**, e.g., $\text{Rep}(D_3),\text{Rep}(S_4)$, etc
* **Fusion rings related to quantum groups**, e.g., $\text{SU}(2)_{k}$ for $k=1,\ldots,6$, $\text{SO}(2N+1)_{2}$ for $N=1,\ldots,3$, etc
* **Fusion rings related to subfactors**, e.g., a ring which we called [Pseudo $\text{PSU}(2)_6$](https://anyonwiki.github.io/pages/FRPages/FR_4_1_2_4.html) (for now)
* ***Extensions*** of the above, such as $\text{HI}(\mathbb{Z}_3)$, $\text{TY}(G)$, for all finite abelian groups $G$ with $|G|<7$
* **Products of the above**, e.g. $\mathbb{Z}_2 \times R$ for any fusion ring $R$ with rank $\leq 3$, etc
* **[Zestings](https://arxiv.org/abs/2005.05544)** of the above
* **Adjoint fusion rings** of rings related to quantum groups, e.g., $\text{Adj}(\text{SO}(11)_2)$, $\text{Adj}(\text{SO}(16)_2)$ 


There are a few fusion rings that I don't immediately recognize as any of the above, though. (Which doesn't mean they aren't of the form above) 

So, I wondered if anyone recognizes some of the fusion rings below as being related to some *known* constructions. They are all categorifiable into a unitary fusion category, but none of the categories admit a braided structure.

1. The [Moore-Read](https://anyonwiki.github.io/pages/FRPages/FR_6_1_4_2.html) fusion ring 

This one is [known from physics](https://www.sciencedirect.com/science/article/pii/055032139190407O/pdf). It has Frobenius-Perron dimensions $(1,1,1,1,\sqrt{2},\sqrt{2})$ and the following multiplication table

$\begin{array}{|llllll|}
\hline
 \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} \\
 \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} \\
 \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{6} & \mathbf{5} \\
 \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{6} & \mathbf{5} \\
 \mathbf{5} & \mathbf{5} & \mathbf{6} & \mathbf{6} & \mathbf{3}+\mathbf{4} & \mathbf{1}+\mathbf{2} \\
 \mathbf{6} & \mathbf{6} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2} & \mathbf{3}+\mathbf{4} \\
\hline
\end{array}$

Is there any way that this ring fits in the groups above?

2. A ring we called [Pseudo $\text{Rep}(S_4)$](https://anyonwiki.github.io/pages/FRPages/FR_5_1_2_4.html)

Its multiplication table looks a lot like that from $\text{Rep}(S_4)$. It is listed in [this paper](https://link.springer.com/content/pdf/10.1007/s11005-022-01542-1.pdf) at page 11 but the paper also says: Model Unknown. Its Frobenius-Perron dimensions are $(1,1,2,3,3)$ and its multiplication table is the following.
 
$\begin{array}{|lllll|}
\hline
 \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} \\
 \mathbf{2} & \mathbf{1} & \mathbf{3} & \mathbf{5} & \mathbf{4} \\
 \mathbf{3} & \mathbf{3} & \mathbf{1}+\mathbf{2}+\mathbf{3} & \mathbf{4}+\mathbf{5} & \mathbf{4}+\mathbf{5} \\
 \mathbf{4} & \mathbf{5} & \mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\
 \mathbf{5} & \mathbf{4} & \mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{2}+\mathbf{3}+\mathbf{4}+\mathbf{5} \\
\hline
\end{array}$

3. A ring called [$\text{FR}^{7,1,2}_{3}$](https://anyonwiki.github.io/pages/FRPages/FR_7_1_2_3.html) of which I have no idea where it comes from.
Its Frobenius-Perron dimensions are $(1,1,1,1,2,2,2)$ and it has the following multiplication table


$\begin{array}{|lllllll|}
\hline
 \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\
 \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\
 \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\
 \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{7} & \mathbf{6} & \mathbf{5} \\
 \mathbf{5} & \mathbf{5} & \mathbf{7} & \mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{6} \\
 \mathbf{6} & \mathbf{6} & \mathbf{6} & \mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{5}+\mathbf{7} \\
 \mathbf{7} & \mathbf{7} & \mathbf{5} & \mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{6} & \mathbf{5}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{6} \\
\hline
\end{array}$

it is related via the testing construction to [this ring](https://anyonwiki.github.io/pages/FRPages/FR_7_1_4_3.html) with the same dimensions but a *reshuffle* of some of its structure constants.

$\begin{array}{|lllllll|}
\hline
 \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\
 \mathbf{2} & \mathbf{1} & \mathbf{4} & \mathbf{3} & \mathbf{5} & \mathbf{6} & \mathbf{7} \\
 \mathbf{3} & \mathbf{4} & \mathbf{2} & \mathbf{1} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\
 \mathbf{4} & \mathbf{3} & \mathbf{1} & \mathbf{2} & \mathbf{5} & \mathbf{7} & \mathbf{6} \\
 \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{3}+\mathbf{4} & \mathbf{6}+\mathbf{7} & \mathbf{6}+\mathbf{7} \\
 \mathbf{6} & \mathbf{6} & \mathbf{7} & \mathbf{7} & \mathbf{6}+\mathbf{7} & \mathbf{3}+\mathbf{4}+\mathbf{5} & \mathbf{1}+\mathbf{2}+\mathbf{5} \\
 \mathbf{7} & \mathbf{7} & \mathbf{6} & \mathbf{6} & \mathbf{6}+\mathbf{7} & \mathbf{1}+\mathbf{2}+\mathbf{5} & \mathbf{3}+\mathbf{4}+\mathbf{5} \\
\hline
\end{array}$