# gr.group theory – List of small groups presented symmetrically non-abelian

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

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

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