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I have a problem where I have groups generated by two $n \times n$ integer matrices and I know that one of those matrices is a permutation matrix.

Question: What is the best way to find minimal nice relations of such a group (knowing that the group is finite)?

I tried Sage and GAP but usually the relations do not look very nice for the human eye although often the group is isomorphic to a symmetric group, so maybe there are nice relations.

Question: Is there a modern way to tackle this problem to find nice relations? Im thinking about some AI, but after trying ChatGPT Im not so optimistic about this.

Here a small example with Sage:

n=2^3
C=matrix([[0,0,0,-1],[1,0,0,-1],[0,1,0,-1],[0,0,1,-1]])
P=matrix([[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]])
display(C)
display(P)
G = MatrixGroup([C,P])
display(len(G))
display(G.as_permutation_group().structure_description())
display(G.as_permutation_group().as_finitely_presented_group(reduced=True))

The result is:

Finitely presented group < a, b | (a*b^-1)^2, a^5, (b*a*b^2)^2, b*a^2*b^2*a^-1*b^-1*a^-1*(b*a)^2, (b*a^2*b)^3 >

and that is isomorphic to the symmetric group $S_5$, but the presentation is quite ugly and I wonder whether there is a computer algebra system (maybe with the help of ai) giving simpler relations (prettier relations for humans).

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  • $\begingroup$ By "finding relations" do you mean "finding a set of relators" (i.e., finding a set of relations generating all other relations)? $\endgroup$ Commented Jan 1 at 23:49
  • $\begingroup$ @YCor Yes, I want to find a minimal set of relations that are as nice as possible. I added an example. $\endgroup$ Commented Jan 2 at 0:21
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    $\begingroup$ Often “pretty relations for humans” are pretty bad for doing anything computational, and vice versa. For example, for computational purposes a very nice type of defining relations are finite complete rewriting systems, but they’re typically quite ugly to humans. $\endgroup$ Commented Jan 2 at 2:52
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    $\begingroup$ (“Pretty” is also seemingly quite subjective since I think your presentation for $S_5$ looks perfectly reasonably pretty, except a little bit of roughness on the fourth relation) $\endgroup$ Commented Jan 2 at 2:53
  • $\begingroup$ @Carl-FredrikNybergBrodda But maybe there could be nicer relations, say powers of words of length <=3. Or is this impossible when the Sage output is that ugly? $\endgroup$ Commented Jan 2 at 10:41

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