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Let $\ G\ $ be a finite group with $\ |G|=n\ $

Caley's theorem states that $\ G\ $ is isomorphic to a subgroup of the symmetric group $\ S_n\ $

However, in many cases, the smallest positive integer $\ m\ $, such that $\ G\ $ is isomorphic to a subgroup of $\ S_m\ $ is smaller than $\ n\ $.

How can I determine $\ m\ $ , if I do not have access to GAP ?

Is there some applet or online calculator that can do this ? Or can $\ m\ $ be calculated by knowing specific properties of $\ G \ $ ?

Peter
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    I believe this is a rather hard problem in general. Perhaps this may be of some use. – morrowmh Apr 27 '20 at 12:52
  • Also https://www.jstor.org/stable/2373739. – Groups Apr 27 '20 at 12:54
  • @Sunyata If I could do this for any group upto, lets say, order $100$ , I would already be content. – Peter Apr 27 '20 at 12:54
  • This answer may also be helpful. – morrowmh Apr 27 '20 at 12:58
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    There's no way you can do this without using a computer. See also this – the_fox Apr 27 '20 at 13:01
  • @the_fox I found two online calculators : "wims permutation groups" and "magma-online-calculator" which are helpful in special cases. Do you know how to use them for more complicated groups ? – Peter Apr 27 '20 at 13:04
  • I am only familiar with GAP, unfortunately. GAP has the capacity to do this calculation, but be a little wary as there was a bug reported. I am not sure if it has since been fixed. – the_fox Apr 27 '20 at 13:07
  • This has been asked a bunch of times, see also https://math.stackexchange.com/questions/36846/minimal-embedding-of-a-group-into-the-group-s-n – verret Apr 28 '20 at 01:35

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