# Perfect group of order 190080

I need to know some properties of the perfect group of order $190080$ which is the Schur cover of the Mathieu group ${\rm M}_{12}$, but when using PerfectGroup(190080), GAP runs so slowly. Is there any other method in GAP for getting this group?

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Just write

gap> G := PerfectGroup(IsPermGroup,190080);
M12 2^1


in order to get the desired group as a permutation group:

gap> GeneratorsOfGroup(G);
[ (3,6)(7,10)(9,12)(13,16)(15,18)(19,22)(20,23)(21,24),
(1,2,3)(4,5,7)(6,8,9)(10,11,13)(12,14,15)(16,17,19)(18,20,21)(22,24,23),
(1,4)(2,5)(3,7)(6,10)(8,11)(9,13)(12,16)(14,17)(15,19)(18,22)(20,24)(21,23)
]


Now computations should be fast. (If you omit the first argument, you get the group as a finitely presented group, and computations with such groups are inefficient for the obvious reasons.)

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Thank you veryyyy much. Your advice was really useful. – Sara Jan 5 at 18:02
One more problem. How I can find the representation of a finite group as a permutation group, because by order GeneratorsOfGroup, generators obtained by a and b .... – Sara Jan 5 at 18:52
@Sara: You can obtain a faithful permutation representation of a finite group $G$ by IsomorphismPermGroup($G$), and the isomorphic permutation group is Image(IsomorphismPermGroup($G$)). – Stefan Kohl Jan 5 at 19:28
Thank you very much. This solve my problem – Sara Jan 5 at 19:42
@Sara It may be worth pointing out that IsomorphismPermGroup is not necessarily of smallest degree. To get a smaller degree permutation representation of a permutation group, use SmallerDegreePermutationRepresentation to get a faithful representation of (potentially) smaller degree. Take the Image as before to get the actual permutation group. This still doesn't guarantee the smallest possible degree representation, but computations tend to be faster in smaller degree so it is helpful to reduce the degree like this. – zibadawa timmy Mar 26 at 23:39