// This file contains a list of 25 permutations $A$ of 0,...,23
// which have the properties
// (i) $A$ has cycle type $3^8$,
// (ii) $AD_2$ has cycle type $2^{12}$, where $D_2$ is the permutation
//      (0,1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23)
// (iii) each $A$ satisfying (i) and (ii) is conjugate by a permutation $C$
//       which commutes with $D_2$ to one of the 25 $A$'s in the list.
// It turns out that there are only 3204 such permutations $A$, and that

G:=SymmetricGroup({0 .. 23});
D2:=G!((0,1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23));

Alist:=[G |
 // 2 As with commutator in Z_G(D2) of size 1
 (0,12,14)(1,13,11)(2,10,18)(3,17,20)(4,19,9)(5,8,22)(6,21,16)(7,15,23),
 (0,12,15)(1,14,19)(2,18,8)(3,7,22)(4,21,10)(5,9,17)(6,16,23)(11,20,13),
// 15 As with commutator in Z_G(D2) of size 2
 (0,2,13)(1,12,14)(3,11,16)(4,15,23)(5,22,18)(6,17,10)(7,9,20)(8,19,21),
 (0,2,20)(1,19,21)(3,11,16)(4,15,23)(5,22,18)(6,17,10)(7,9,13)(8,12,14),
 (0,2,13)(1,12,14)(3,11,7)(4,6,9)(5,8,10)(15,23,19)(16,18,21)(17,20,22),
 (0,2,13)(1,12,14)(3,11,19)(4,18,9)(5,8,22)(6,21,16)(7,15,23)(10,17,20),
 (0,2,13)(1,12,14)(3,11,19)(4,18,21)(5,20,10)(6,9,16)(7,15,23)(8,22,17),
 (0,3,20)(1,19,6)(2,5,21)(4,11,22)(7,18,13)(8,12,15)(9,14,17)(10,16,23),
 (0,3,21)(1,20,7)(2,6,22)(4,11,17)(5,16,23)(8,19,13)(9,12,15)(10,14,18),
 (0,3,18)(1,17,9)(2,8,19)(4,11,22)(5,21,13)(6,12,15)(7,14,20)(10,16,23),
 (0,17,13)(1,12,5)(2,4,8)(3,7,9)(6,23,10)(11,22,18)(14,16,20)(15,19,21),
 (0,4,22)(1,21,8)(2,7,18)(3,17,23)(5,11,15)(6,14,19)(9,20,13)(10,12,16),
 (0,4,19)(1,18,9)(2,8,15)(3,14,20)(5,11,22)(6,21,13)(7,12,16)(10,17,23),
 (0,4,18)(1,17,23)(2,22,9)(3,8,19)(5,11,13)(6,12,16)(7,15,20)(10,21,14),
 (0,4,20)(1,19,6)(2,5,11)(3,10,21)(7,18,13)(8,12,16)(9,15,22)(14,17,23),
 (0,23,18)(1,17,8)(2,7,10)(3,9,16)(4,15,21)(5,20,13)(6,12,11)(14,19,22),
 (0,18,13)(1,12,6)(2,5,10)(3,9,20)(4,19,11)(7,23,16)(8,15,21)(14,17,22),
 // 4 As with commutator in Z_G(D2) of size 4
 (0,2,13)(1,12,14)(3,11,16)(4,15,23)(5,22,9)(6,8,19)(7,18,20)(10,21,17),
 (0,2,13)(1,12,14)(3,11,22)(4,21,17)(5,16,9)(6,8,19)(7,18,20)(10,15,23),
 (0,17,13)(1,12,5)(2,4,9)(3,8,10)(6,23,19)(7,18,11)(14,16,21)(15,20,22),
 (0,5,22)(1,21,8)(2,7,15)(3,14,19)(4,18,23)(6,11,16)(9,20,13)(10,12,17),
 // 3 As with commutator in Z_G(D2) of size 6
 (0,2,13)(1,12,14)(3,11,7)(4,6,17)(5,16,18)(8,10,21)(9,20,22)(15,23,19),
 (0,2,13)(1,12,14)(3,11,7)(4,6,21)(5,20,22)(8,10,17)(9,16,18)(15,23,19),
 (0,19,13)(1,12,7)(2,6,10)(3,9,16)(4,15,21)(5,20,11)(8,23,17)(14,18,22),
 // one As with commutator in Z_G(D2) of size 8
 (0,5,13)(1,12,17)(2,16,9)(3,8,22)(4,21,14)(6,11,19)(7,18,23)(10,15,20)];

Z24:=RSpace(IntegerRing(),24);
Aseqlist:=[Z24 |
[12,13,10,17,19,8,21,15,22,4,18,1,14,11,0,23,6,20,2,9,3,16,5,7],
[12,14,18,7,21,9,16,22,2,17,4,20,15,11,19,0,23,5,8,1,13,10,3,6],
[2,12,13,11,15,22,17,9,19,20,6,16,14,0,1,23,3,10,5,21,7,8,18,4],
[2,19,20,11,15,22,17,9,12,13,6,16,14,7,8,23,3,10,5,21,0,1,18,4],
[2,12,13,11,6,8,9,3,10,4,5,7,14,0,1,23,18,20,21,15,22,16,17,19],
[2,12,13,11,18,8,21,15,22,4,17,19,14,0,1,23,6,20,9,3,10,16,5,7],
[2,12,13,11,18,20,9,15,22,16,5,19,14,0,1,23,6,8,21,3,10,4,17,7],
[3,19,5,20,11,21,1,18,12,14,16,22,15,7,17,8,23,9,13,6,0,2,4,10],
[3,20,6,21,11,16,22,1,19,12,14,17,15,8,18,9,23,4,10,13,7,0,2,5],
[3,17,8,18,11,21,12,14,19,1,16,22,15,5,20,6,23,9,0,2,7,13,4,10],
[17,12,4,7,8,1,23,9,2,3,6,22,5,0,16,19,20,13,11,21,14,15,18,10],
[4,21,7,17,22,11,14,18,1,20,12,15,16,9,19,5,10,23,2,6,13,8,0,3],
[4,18,8,14,19,11,21,12,15,1,17,22,16,6,20,2,7,23,9,0,3,13,5,10],
[4,17,22,8,18,11,12,15,19,2,21,13,16,5,10,20,6,23,0,3,7,14,9,1],
[4,19,5,10,20,11,1,18,12,15,21,2,16,7,17,22,8,23,13,6,0,3,9,14],
[23,17,7,9,15,20,12,10,1,16,2,6,11,5,19,21,3,8,0,22,13,4,14,18],
[18,12,5,9,19,10,1,23,15,20,2,4,6,0,17,21,7,22,13,11,3,8,14,16],
[2,12,13,11,15,22,8,18,19,5,21,16,14,0,1,23,3,10,20,6,7,17,9,4],
[2,12,13,11,21,16,8,18,19,5,15,22,14,0,1,23,9,4,20,6,7,17,3,10],
[17,12,4,8,9,1,23,18,10,2,3,7,5,0,16,20,21,13,11,6,22,14,15,19],
[5,21,7,14,18,22,11,15,1,20,12,16,17,9,19,2,6,10,23,3,13,8,0,4],
[2,12,13,11,6,16,17,3,10,20,21,7,14,0,1,23,18,4,5,15,22,8,9,19],
[2,12,13,11,6,20,21,3,10,16,17,7,14,0,1,23,18,8,9,15,22,4,5,19],
[19,12,6,9,15,20,10,1,23,16,2,5,7,0,18,21,3,8,22,13,11,4,14,17],
[5,12,16,8,21,13,11,18,22,2,15,19,17,0,4,20,9,1,23,6,10,14,3,7]];

for ii in [1 .. #Aseqlist] do
 if Alist[ii] ne G!(ElementToSequence(Aseqlist[ii])) then print "error"; end if;
end for;

Cgen1:=G!((0,12)(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23));
Cgen2:=G!((0,1,2,3,4,5,6,7,8,9,10,11));
// The following is the centralizer of $D2$:
cC:=sub< G | Cgen1,Cgen2>;

centralizercountlist:=[IntegerRing() | ];
for ii in [1 .. #Alist] do
 centralizercountlist:=Append(centralizercountlist,-1);
end for;

for ii in [1 .. #Alist] do
 perm:=Alist[ii];
 count:=0;
 for C in cC do
  if (perm*C eq C*perm) then count+:=1; end if;
 end for;
 centralizercountlist[ii]:=count;
end for;
print centralizercountlist;