61.27 Examples of DCE and Symmetric Presentations

M_{12}

The following input gives a symmetric presentation of the Mathieu group M_{12}:

```    gap> t := AbstractGenerator("t");;
gap> K := Group((1,2,3,4,5),(1,2,3));
Group( (1,2,3,4,5), (1,2,3) )
gap> SGrec := SetupSymmetricPresentation(K,t);
rec(
skeleton := rec(
groupK := Group( (1,2,3,4,5), (1,2,3) ),
gainGroups := [ rec(
dom := 1,
op := function (...) internal; end ) ],
gens := [ rec(
name := t,
wgg := 1 ) ] ),
makeGen := function ( pt ) ... end )
gap> t := SGrec.makeGen;
function ( pt ) ... end
gap> Pres := SGrec.skeleton;
rec(
groupK := Group( (1,2,3,4,5), (1,2,3) ),
gainGroups := [ rec(
dom := 1,
op := function (...) internal; end ) ],
gens := [ rec(
name := t,
wgg := 1 ) ] )
gap> Pres.name := "M12 Symmetric";
"M12 Symmetric"
gap> Pres.strategy := rec(EC := [1000..3000]);
rec(
EC := [ 1000 .. 3000 ] )
gap> Pres.relators := [t(1)^3,(t(1)/t(2))^2*DCEWord(K,(3,4,5))];
[ DCEWord(Group( (1,2,3,4,5), (1,2,3) ),[t])^3,
DCEWord(Group( (1,2,3,4,5),
(1,2,3) ),[t, (1,3,4,5,2), t^-1, (1,2,5,4,3), t, (1,3,4,5,2), t^-1\
, (1,2,5,4,3), (3,4,5)]) ]
gap> Pres.subgens := [DCEWord(K,(1,2,3,4,5)),DCEWord(K,(1,2,3)),
>                  (DCEWord(K,(1,2,3,4,5))*t(1))^8];
[ DCEWord(Group( (1,2,3,4,5), (1,2,3) ),[(1,2,3,4,5)]),
DCEWord(Group( (1,2,3,4,5), (1,2,3) ),[(1,2,3)]),
DCEWord(Group( (1,2,3,4,5), (1,2,3) ),[(1,2,3,4,5), t])^8 ]
gap> Pres.relators[1].weight := 2;;   # default weight is too low```

DCE enumerates this presentation in a few seconds.

```    gap> InfoDCE1 := Ignore;
function (...) internal; end
gap> u := DCE(Pres);
<< Double coset table "M12 Symmetric" early-closed 47 double
1584 single >>
gap> time;
5400```

He: 2

The following is a presentation of He: 2 generated by 180 symmetric generators of order 7 permuted by 3S_7times 2. This is really 30 generators permuted monomially, but we don't have monomial groups in GAP.

The following can be placed in an input file `he2.g`.

```#
# The group K we want is 3S7 x 2. We make this from a handy
# representation of 3S7
#
DoubleP := function(p,n)
local l;
l := OnTuples([1..n],p);
Append(l,l+n);
return PermList(l);
end;

Swap := function(n)
return PermList(Concatenation([n+1..2*n],[1..n]));
end;

K := Group(
DoubleP((1, 2)( 3, 5)( 4, 7)( 6,10)( 8,12)( 9,14)(11,17)(13,20)
(15,23)(16,25)(18,28)(19,30)(21,33)(22,35)(24,37)(26,40)(27,41)
(29,44)(31,47)(32,49)(34,51)(36,54)(38,57)(39,46)(42,61)(43,63)
(45,66)(48,53)(50,70)(52,60)(55,73)(56,65)(58,76)(59,78)(62,75)
(64,80)(67,84)(68,74)(69,77)(71,85)(72,86)(79,89)(81,88)(82,87)
(83,90),90),
DoubleP(( 1, 3, 6)(44,65,49)
(2,4,8,13,21,34,52,10,16,26,28,43,64,82,5,9,15,24,38,58,77)
(7,11,18,29,45,67,63,14,22,20,32,50,61,33,25,39,37,56,75,86,57)
(12,19,31,48,69,51,71,23,36,55,74,87,76,88,40,59,79,41,60,80,90)
(17,27,42,62,81,47,30,46,68,84,70,85,89,78,35,53,72,66,83,73,54)
,90),Swap(90) );
#
# Now lets get the generators we want
#
x := DCEWord(K,K.1);
y := DCEWord(K,K.2);
a := DCEWord(K,K.3);
#
# And the name for our generator outside K
#
t := AbstractGenerator("t");
#
# Now we can specify our setup
#
SGrec := SetupSymmetricPresentation(K,t);
SG := SGrec.makeGen;
Pres := SGrec.skeleton;
#
# We still have to put some fields in the presentation
#
Pres.name := "He:2 Symmetric";
Pres.relators := [
SG(1)^7,(SG(1)* SG(2))^2,
SG(1)^2 / SG(3),
y^-7 / (SG(1)^-1*SG(2)^-2*SG(1)^2*SG(2)),
y^9 /  Comm(SG(1),SG(65)),
SG(1)*SG(91),
DCEWord(K,DoubleP((1,2)(3,5)(4,76)(6,10)(7,58)(8,12)(9,80)(11,70)
(13,20)(14,64)(15,23)(16,51)(17,50)(18,28)(19,42)(21,87)
(22,62)(24,37)(25,34)(26,40)(27,32)(29,68)(30,61)(31,85)
(33,82)(35,75)(36,72)(38,60)(39,66)(41,49)(43,69)(44,74)
(45,46)(47,71)(48,65)(52,57)(53,56)(54,86)(55,81)(59,84)
(63,77)(67,78)(73,88)(79,83)(89,90),90)) /
(SG(1)*SG(2)^2*SG(1)^2*SG(2)) ];
Pres.subgens := [t,x,x^(y^3)*x^(y^-1*x*y^-2),
Comm(x,y^-1*x*y^-1),Comm(x,y*x*y^2),a];
Pres.strategy := rec(EC := [8000..12000]);```

We can run this example quietly:

```    gap> Read("he2.g");
gap> InfoDCE1 := Ignore;
function (...) internal; end
gap> u := DCE(Pres);
<< Double coset table "He:2 Symmetric" early-closed 9 double
8330 single >>
gap> time;
126716```

GAP 3.4.4
April 1997