gap> h1:=(1,12)(2,11)(3,10)(4,9)(5,8)(6,7);
( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)
gap> h2:=(2,3,5,9,8,10,6,11,4,7,12);
( 2, 3, 5, 9, 8,10, 6,11, 4, 7,12)
gap> IsSimple(H);
true
gap> Size(H);
95040
gap> M12 := MathieuGroup( 12 );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11), ( 3, 7,11, 8)( 4,10, 5, 6), ( 1,12)
( 2,11)( 3, 6)( 4, 8)( 5, 9)( 7,10) )
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> Size(M12);
95040
gap> Size(M24);
244823040
gap> F2:= FreeGroup( "a", "b");
Group( a, b )
gap> words := [ F2.1, F2.2 ];
[ a, b ]
gap> P := PresentationViaCosetTable(H,F2,words);
<< presentation with 2 gens and 6 rels of total length 77 >>
gap> G := FpGroupPresentation( P );
Group( f.1, f.2 )
gap> G.relators;
[ f.1^2, f.1*f.2*f.1*f.2^-1*f.1*f.2*f.1*f.2^-1, f.2^11, 
  f.1*f.2^2*f.1*f.2^-2*f.1*f.2^2*f.1*f.2^-2, 
  f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*f.2*f.1*\
f.2, 
  f.1*f.2*f.1*f.2^2*f.1*f.2^2*f.1*f.2*f.1*f.2^2*f.1*f.2^2*f.1*f.2*f.1*f.2^2*f.\
1*f.2^2 ]
gap> SimplifyPresentation( P );
#I  there are 2 generators and 6 relators of total length 77
gap> quit;

#
# M12 is <a,b | a^2=1, b^11=1, (ab^2ab^-2)^2=1, (ab)^9=1, 
#               (ab(ab^2)^2)^3=1, (abab^-1)^2=1 >
#
#
#