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.