Let $F_n$ be the free group generated by $x_1,\ldots,x_n$ and let $S_n$ be the symmetric group on $\{1,\cdots,n\}$. Let $w=x_{i_1}^{\pm1}\cdots x_{i_s}^{\pm1}$ be a word and for each $\sigma \in S_n$, define $\sigma(w)=x_{\sigma(i_1)}^{\pm1}\cdots x_{\sigma(i_s)}^{\pm1}$. We consider groups of the form

$$G_n(w)=\langle x_1,\ldots,x_n\mid\sigma(w), \sigma\in S_n\rangle,$$ where $w$ is a given word in $F_n$. Such groups are called symmetrically presented. For example, it can be proven that $$G_4(x_1x_2^2x_3x_4^{-1})=\langle x_1,x_2,x_3,x_4\mid\sigma(x_1x_2^2x_3x_4^{-1}), \sigma\in S_n\rangle$$is a non-Abelian group of order $96$.

My question is, given $n$, what is the smallest non-Abelian symmetrically presented group? Any list of examples of non-Abelian symmetrically presented groups will also be much appreciated.

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    $\begingroup$ $\langle x_1,x_2\mid x_1^{-1}x_2x_1x_2,x_2^{-1}x_1x_2x_1\rangle$ is the quaternion group, of order $8$. $\endgroup$ – verret Mar 26 at 21:29
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    $\begingroup$ And $S_3\cong\langle x_1,x_2,x_3\mid \sigma (x_1x_2x_3x_2^{-1}), \sigma\in S_3\rangle$, so that's the smallest non-abelian example. $\endgroup$ – verret Mar 26 at 21:34
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    $\begingroup$ Robert Curtis has studied symmetric presentations of some sporadic finite simple groups. $\endgroup$ – Richard Lyons Mar 26 at 23:13

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