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I fond a lot of examples using presentation of $D_8$ by generators which are permutations of $S_4$.

1) How many presentations could be found?

2) Could it be presented by permutations of $S_5$ or any different group?

Thanks for advance

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    $\begingroup$ What counts as different presentations? $\endgroup$ – Tobias Kildetoft Dec 5 '18 at 10:36
  • $\begingroup$ @TobiasKildetoft they don´t look the same! $\endgroup$ – A.Messab Dec 5 '18 at 10:39
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    $\begingroup$ Then there will be infinitely many presentations, since I can just keep adding new relations. $\endgroup$ – Tobias Kildetoft Dec 5 '18 at 10:40
  • $\begingroup$ @TobiasKildetoft It´s not like one is equipped with more relationd than the other, it´s about seeing a different sets of generators plus different sets of relators, and by different I mean looks different not just sized differently $\endgroup$ – A.Messab Dec 5 '18 at 10:44
  • $\begingroup$ I think you may want to look up the definition of a presentation, because it does mean what you seem to be using it to mean here. $\endgroup$ – Tobias Kildetoft Dec 5 '18 at 10:47

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