My question is:
Is there a classification of finite groups representable as a $\mathbb{K}$-reflection group for some division ring $\mathbb{K}$ of characteristic zero?
I would also appreciate any references dealing with this problem or with particular cases. If the classification turns out to be intractable, I would ask if at least an example can be found of a new exceptional reflection group of rank $\ge 3$ (i.e. not part of an infinite family, see details below).
Details
Let $\mathbb{K}$ be a division ring of characteristic zero, and $V$ a right vector space of dimension $n$ over $\mathbb{K}$. A finite $\mathbb{K}$-reflection group of rank $n$ can be defined as a finite group $G$ of linear transformations in $V$ generated by elements of finite order that leave a hyperplane pointwise fixed (we consider only the case where $G$ is essential, i.e. no nontrivial subspace is fixed by the whole of $G$).
Fields
The irreducible finite reflection groups over $\mathbb{C}$ (also called unitary or complex reflection groups) were classified by Shephard and Todd. The classification consists essentially of:
An infinite family $G(m,p,n)$ of groups of rank $n$, where $p$ divides $m$, with slightly different properties when $m=p=1$ or when $n=1$ (these two cases are almost always listed independently). This family includes the familiar Coxeter groups $A_n, B_n, D_n, I_2(m)$ as special cases.
34 exceptional cases of rank $\le 8$. These include the exceptional Coxeter groups $E_6, E_7, E_8, F_4, H_3, H_4$.
If $\mathbb{K}$ is a field of characteristic $0$, it is known (see e.g. Section 15-2 here) that if a group has a representation as a $\mathbb{K}$-reflection group then it also has a representation as a complex reflection group, so we get no new examples for fields.
The quaternions
I'm interested in the case where $\mathbb{K}$ is not a field. I have found no information on this case except when $\mathbb{K}=\mathbb{H}$, the skew-field of quaternions. The irreducible finite quaternionic reflection groups (also known under another form as symplectic reflection groups, e.g. here) were classified in this paper by Cohen. Essentially:
There is an infinite family $G_n(M,P,\alpha)$ of groups of rank $n$, where $M$ is a finite subgroup of $\mathbb{H}$, $P$ is a normal subgroup of $M$ and some conditions are satisfied, with an extra datum $\alpha$ (an automorphism of $M/P$ of order $\le 2$) in the case $n=2$. If $M, P$ are cyclic groups we recover the infinite family of complex reflection groups from the previous list.
There is an additional family $E(H)$ of groups of rank $2$.
There are 13 new exceptional cases of rank $\le 5$, including a double cover of the Hall-Janko group $HJ$, a sporadic finite simple group, in rank $3$.
Other division rings?
Unlike the case for fields, not all possible finite reflection groups for division rings of characteristic zero can be found in the above list. For example, the "baby problem" of classifying finite $\mathbb{K}$-reflection groups of rank $1$ (equivalently the possible finite subgroups of division rings of characteristic $0$) was dealt with in this paper by Amitsur; there are additional groups not realised in the quaternions.
Nevertheless, I'm hopeful that it is still possible to find a full list for all $\mathbb{K}$, perhaps using a technique similar to the case of fields to restrict our analysis to division algebras of finite dimension over $\mathbb{Q}$ (which are always cyclic algebras over their center $Z(\mathbb{K})$, a finite extension of $\mathbb{Q}$).
In view of the previous lists, I would expect a full classification (if there is one) to follow a similar form:
An infinite family $G_n(M,P,\alpha)$ of groups of rank $n$, where $M$ is a finite subgroup of a division ring $\mathbb{K}$, $P \trianglelefteq M$ and some conditions are satisfied, possibly with some extra data $\alpha$ in low rank.
A family or families of examples in rank $2$. I'm not very interested in these, a brute-force method to find them might be to start with the classification of finite subgroups of $GL(2,\mathbb{K})$ and perform a tedious case-by-case check.
A number of exceptional cases of small rank. These are the ones I'm most interested in. In the quaternionic case these are precisely the primitive reflection groups whose complexification is also primitive; in the general case they could correspond to primitive reflection groups which remain primitive after tensoring with a splitting field (this is just a guess).
