Kthulhu racks

In [AFGV], [ACG1], [ACG3] the notions of racks of type D, F and C were introduced.  

Since we deal with subracks of conjugacy classes, we recall the translation of these notions in this case.

Definition. Let $\mathcal O$ be a conjugacy class in a finite group $M$.

$\mathcal O$ is of type C if there exists $H \leq M$ and $r,s \in H$ such that
$rs \neq sr$, $\mathcal{O}_r^H \neq\mathcal{O}_s^H$, $H=\langle \mathcal{O}_r^H, \mathcal{O}_s^H\rangle$
and either $\min\{\vert \mathcal O_r^H\vert, \,\vert \mathcal O_s^H\vert\}\geq 2$ or $\max \{\vert\mathcal O_r^H\vert, \,\vert\mathcal O_s^H\vert\}\geq 4$.

$\mathcal O$ is of type D if there exists $r, s \in M$ such that $(rs)^2 \neq (sr)^2$,
$\mathcal O_r^{\langle r,s \rangle} \neq\mathcal O_s^{\langle r,s \rangle}.$

$\mathcal O$ is of type F if there exists $r_a\in \mathcal O$, $a\in \mathbb I_4 = \{1,2,3,4\}$, such that $r_a r_b \neq r_br_a$, $a \neq b\in \mathbb I_4$.

Theorem. [AFGV, Theorem 3.6], [ACG1, Theorem 2.8], [ACG3, Theorem 2.9]. A conjugacy class of type D, F or C collapses, i.e. the dimension of the Nichols algebra $\mathcal B(\mathcal O, q)$ is infinite for any cocycle $q$.
A rack is kthulhu if it is not of type C, D or F. That is, the kthulhu conjugacy classes are those that might support a finite-dimensional Nichols algebra, because they are not discarded by the Theorem. In other words, for a kthuhlu rack $X$ we do need

• to compute all cocycles $q\in H^2(X, \mathbf{GL}_n(\mathbb C)$,
• to compute the Nichols algebra $\mathcal B(X, q)$ for every such $q$.

Since any indecomposable rack $Y$ always admits a rack epimorphism onto a simple rack $Y$, the following proposition tells us that it sufficies to analize simple racks.

Proposition. [AFGV1, ACG1, ACG3]
Let $X \to Y$ be a surjective morphism of racks. If $Y$ is not kthulhu, then $X$ is not kthulhu.

In conclusion, we arrive at the next problem:

Problem
Determine all simple racks that are not kthulhu.