Let $G = \left\langle S | R \right\rangle$ be a finitely presented group where S is a set of generators and R is a set of relations. We say that the presentation is "locally commuting" if whenever two generators $a, b$ appear in a word in R, the word $aba^{-1}b^{-1}$ also belongs to R. Following Slofstra, I'll call a group which admits a locally commuting presentation a "solution group".

**Problem:**
Which finite nonabelian groups are solution groups?

(Slofstra showed that every finitely-presented group embeds into a solution group. However, the methods used there seem too coarse for this problem. In particular, he gives a construction which takes finitely-presented G and outputs a solution group G' and embedding G -> G', but if we feed different presentations of the same group in for G we can get out different groups for G'.)

I know of one infinite family of nonabelian finite solution groups. Quantum computing theorists know these as the $n$-qubit Pauli groups for $n\geq 2$. They are also known as the Heisenberg groups over $\mathbb F_2$. These are the $(n+2)\times (n+2)$ upper-triangular matrices with the following structure: $$ \begin{pmatrix} 1 & a & c \\ 0 & I_n & b \\ 0 & 0 & 1 \end{pmatrix},\quad a,b, \in \mathbb F_2^n, c \in \mathbb F_2. $$ There are two cute constructions for $n=2$ and $n=3$. Larger $n$ can be formed by appropriate products.

**Problem, restated:** Are there any nonabelian finite solution groups other than the Pauli groups on $n$-qubits, $n \geq 2$?

Here I'll show the cute constructions. This paper with Andrea Coladangelo has picture-proofs that these presentations give the right groups. (The results there stated for general $d$ are false, but still hold when $d=2$.)

For $n=2$, we have the Mermin--Peres "Magic Square". $G = \left\langle S | R_0 \cup R_1 \cup R_2 \cup R_\text{comm} \right \rangle$, where

$S = \left\{J,e_1,\ldots,e_9\right\}$ $R_0 = \left\{[s, J] | s\in S \right\},$ $R_1 = \left\{s^2 | s \in S \right\},$ $R_2 = \left\{e_1e_2e_3, e_4e_5e_6, e_7e_8e_9, e_1e_4e_7, e_2e_5e_8, Je_3e_6e_9\right\},$ $R_\text{comm} = \left\{[e_i,e_j] | \text{$e_i$ and $e_j$ appear together in some relation of $R_2$} \right\}$.

(Here $[x,y] := xyx^{-1}y^{-1}$ denotes the group commutator.)

For $n=3$, we have the Mermin--Peres "Magic Pentagram". $G = \left\langle S | R_0 \cup R_1 \cup R_2 \cup R_\text{comm} \right \rangle$, where

$S = \left\{J,e_1,\ldots,e_{10}\right\}$ $R_0 = \left\{[s, J] | s \in S \right\},$ $R_1 = \left\{s^2 | s \in S \right\},$ $R_2 = \left\{e_1e_2e_8e_9, e_2e_3e_6e_7, e_3e_4e_9e_{10}, e_4e_5e_7e_8, Je_5e_6e_{10}e_1\right\},$ $R_\text{comm} = \left\{[e_i,e_j] | \text{$e_i$ and $e_j$ appear together in some relation of $R_2$} \right\}$