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.