# List of Small Non-Abelian Symmetrically Presented Groups

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.

• $\langle x_1,x_2\mid x_1^{-1}x_2x_1x_2,x_2^{-1}x_1x_2x_1\rangle$ is the quaternion group, of order $8$. – verret Mar 26 at 21:29
• 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. – verret Mar 26 at 21:34
• Robert Curtis has studied symmetric presentations of some sporadic finite simple groups. – Richard Lyons Mar 26 at 23:13