Sylow $2$-subgroup Mathieu Group $M_{24}$

Question

I need to compute the Sylow $2$-subgroup of the Mathieu Group $M_{24}$. Unfortunately, this is hard to identify with a machine as it is of order $2^{10}$ and therefore not on the GAP library. I have also looked at a few books and papers on the Mathieu group, but could not find the information. Any other suggestions?

Answer 1

While I would emphasize Derek Holt's remark on that "identification" does not make much sense, here is a way that shows the Sylow subgroup to be isomorphic to the semidirect product of an elementary abelian group with a group given by a matrix action: $C_2^6\rtimes (D_8\times C_2)$:

gap> mats:=[[[1,0,0,0,0,0],[1,1,0,0,0,0],[0,0,1,0,0,0],
> [0,0,0,1,0,0],[0,0,1,0,1,0],[0,0,0,1,0,1]],
> [[1,0,0,0,0,0],[0,1,0,0,0,0],[1,0,1,0,0,0],
> [0,0,0,1,0,0],[0,1,0,0,1,0],[0,0,0,0,0,1]],
> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],
> [0,0,1,1,0,0],[0,0,0,0,1,0],[0,0,0,0,1,1]]]*Z(2);;
gap> m:=Group(mats);;
gap> d:=DirectProduct(DihedralGroup(8),CyclicGroup(2));;
gap> IsomorphismGroups(d,m);
[ f1, f2, f3, f4 ] -> [ <an immutable 6x6 matrix over GF2>,
  <an immutable 6x6 matrix over GF2>, <an immutable 6x6 matrix over GF2>,
  <an immutable 6x6 matrix over GF2> ]

This shows the group m is indeed $D_8\times C_2$. Next form the semidirect product and the Sylow subgroup, and show they are isomorphic:

gap> p:=SemidirectProduct(m,GF(2)^6);
<matrix group of size 1024 with 9 generators>
gap> s:=SylowSubgroup(MathieuGroup(24),2);;
ap> IsomorphismGroups(s,p);
[ (1,17)(2,7,12,15)(3,20,22,21)(4,18)(6,23,19,8)(9,13,10,14),
  (1,7,8,10,17,13,23,12)(2,11,14,6,9,5,15,19)(3,20,22,21)(16,24),
  (1,6)(2,14)(3,24)(4,21)(5,8)(7,10)(9,15)(11,23)(12,13)(16,22)(17,19)(18,20)
    , (1,12,17,10)(2,11,9,5)(4,16)(6,15,19,14)(7,23,13,8)(18,24) ] ->
[ <an immutable 7x7 matrix over GF2>, <an immutable 7x7 matrix over GF2>,
  <an immutable 7x7 matrix over GF2>, <an immutable 7x7 matrix over GF2> ]

Answer 2

Maybe there is a beautiful presentation of $P\in Syl_2(G)$, but Magma chucked a few out and I chose the smallest. It is generated by four elements, $a,b,c,d$, subject to the 12 relations $$a^4 = b^4 = c^2 = d^2 = 1,$$ $$(acb^{-1})^2=(ba)^4=1,$$ and then some worse ones: $$ada^{-1}b^2d=a^{-2}b^{-2}a^2b^2=cbcbdb=ca^{-2}dca^2d=caba^{-1}b^2cb=ba^{-2}ca^{-1}b^{-1}a^{-1}c=1.$$ Double-check that that works, as I have just copied it out from Magma's output.

Edit: Best I have so far is $$ a^4=b^4=c^4=d^4=1,$$ $$(ba)^2=(ca)^2=(bc^{-1})^2=(db)^2=(dc)^2=1$$ $$ada^{-1}c^2d^{-1}=d^{-1}b^{-2}a^{-1}da^{-1}=cab^{-1}acb=1.$$

Answer 3

You can do this in GAP as follows:

gap> M24:=MathieuGroup(24);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), 
  (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16), (1,24)
  (2,23)(3,12)(4,16)(5,18)(6,10)(7,20)(8,14)(9,21)(11,17)(13,22)(15,19) 
 ])
gap> S:=SylowSubgroup(M24,2);
<permutation group of size 1024 with 10 generators>
gap> GeneratorsOfGroup(S);
[ (4,19)(5,23)(6,7)(11,15)(12,20)(13,16)(17,24)(21,22), 
  (3,14)(6,12)(7,20)(8,10)(11,16)(13,15)(17,24)(21,22), 
  (1,7,9,6)(2,20,18,12)(3,4,8,19)(5,14,23,10)(11,24)(13,21)(15,17)(16,22)
    , (1,2)(3,10)(4,23)(5,19)(6,20)(7,12)(8,14)(9,18), 
  (3,6)(7,8)(10,20)(11,22)(12,14)(13,17)(15,24)(16,21), 
  (2,5,18,23)(3,7,8,6)(4,19)(10,14)(11,15,22,21)(13,16,17,24), 
  (1,19,9,4)(2,18)(6,7)(10,12,14,20)(11,21,22,15)(13,24,17,16), 
  (3,8)(6,7)(10,14)(11,13)(12,20)(15,16)(17,22)(21,24), 
  (2,18)(3,8)(5,23)(6,7)(11,22)(13,17)(15,21)(16,24), 
  (1,9)(4,19)(10,14)(11,22)(12,20)(13,17)(15,21)(16,24) ]

Enter ?Mathieu in GAP to see the documentation about this and other group constructors.

If by "identify" you mean having another group of order 1024 and checking whether it's isomorphic, you can try IsomorphismGroups:

gap> D:=DihedralGroup(1024);
<pc group of size 1024 with 10 generators>
gap> IsomorphismGroups(S,D);
fail
gap> T:=Group(GeneratorsOfGroup(S));
<permutation group with 10 generators>
gap> IsomorphismGroups(S,T);
[ (4,19)(5,23)(6,7)(11,15)(12,20)(13,16)(17,24)(21,22), 
  (3,14)(6,12)(7,20)(8,10)(11,16)(13,15)(17,24)(21,22), 
  (1,7,9,6)(2,20,18,12)(3,4,8,19)(5,14,23,10)(11,24)(13,21)(15,17)(16,22)
    , (1,2)(3,10)(4,23)(5,19)(6,20)(7,12)(8,14)(9,18), 
  (3,6)(7,8)(10,20)(11,22)(12,14)(13,17)(15,24)(16,21), 
  (2,5,18,23)(3,7,8,6)(4,19)(10,14)(11,15,22,21)(13,16,17,24), 
  (1,19,9,4)(2,18)(6,7)(10,12,14,20)(11,21,22,15)(13,24,17,16), 
  (3,8)(6,7)(10,14)(11,13)(12,20)(15,16)(17,22)(21,24), 
  (2,18)(3,8)(5,23)(6,7)(11,22)(13,17)(15,21)(16,24), 
  (1,9)(4,19)(10,14)(11,22)(12,20)(13,17)(15,21)(16,24) ] -> 
[ (1,6)(2,12)(3,19)(4,8)(5,14)(7,9)(10,23)(11,17)(13,22)(15,24)(16,
    21)(18,20), (2,23)(5,18)(10,12)(11,22)(13,17)(14,20)(15,16)(21,24), 
  (1,7,9,6)(2,20,18,12)(3,4,8,19)(5,14,23,10)(11,24)(13,21)(15,17)(16,22)
    , (1,4)(2,5)(3,7)(6,8)(9,19)(10,20)(11,17)(12,14)(13,22)(15,24)(16,
    21)(18,23), (1,4)(2,23)(5,18)(9,19)(11,17)(13,22)(15,21)(16,24), 
  (1,8,23,12)(2,10,4,6)(3,5,20,9)(7,18,14,19)(11,15,13,16)(17,21,22,24), 
  (1,12,5,3)(2,6,19,14)(4,10,18,7)(8,9,20,23)(11,15,13,16)(17,21,22,24), 
  (1,4)(2,23)(3,7)(5,18)(6,8)(9,19)(10,12)(14,20), 
  (1,23)(2,4)(3,20)(5,9)(6,10)(7,14)(8,12)(11,13)(15,16)(17,22)(18,
    19)(21,24), (1,5)(2,19)(3,12)(4,18)(6,14)(7,10)(8,20)(9,23)(11,
    13)(15,16)(17,22)(21,24) ]