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for a group G, does there exist a group presentation which is the most "symmetric" by which I mean has the most automorphisms of the group by permuting generators?

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If you look up symmetric presentations of groups you will find some articles that may be of interest to you. – Andreas Caranti Feb 20 at 13:39
Well, you could give a presentation for $G$ by having the generators be all the (non-trivial) group elements, and the relations being how they multiply together. For example, the Klein $4$-group is $\langle a, b, c; ab=c, ba=c, bc=a, cb=a, ac=b, ca=b, a^2=1, b^2=1, c^2=1\rangle$. Then, all automorphisms permute the generators. – user1729 Feb 20 at 13:40

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