gap> r1:=(1,2,3,4,5,6,7,8,9,10,11,12);
( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)
gap> r2:=(1,13,14,15,16,17,7,18,19,20,21,22);
( 1,13,14,15,16,17, 7,18,19,20,21,22)
gap> r3:=(4,23,24,15,25,26,10,27,28,20,29,30);
( 4,23,24,15,25,26,10,27,28,20,29,30)
gap> c1:=r1*r3*r1^-1*r3^-1;
( 3, 4,30)( 9,10,26)
gap> c1*c1;
( 3,30, 4)( 9,26,10)
gap> c2:=r1*r2*r1^-1*r2^-1;
( 1,22,12)( 6, 7,17)
gap> #3-cycle around NP, 3-cycle around SP
gap> c3:=r1^6*r2^6*r1^-6*r2^-6;
()
gap> c3:=r1^6*r2^3*r1^-6*r2^-3;
( 1, 7)(15,20)
gap> c4:=r1^3*r2^6*r1^-3*r2^-6;
( 1, 7)( 4,10)
gap> c4:=r1^3*r2^3*r1^-3*r2^-3;
( 1,20,10)( 4, 7,15)
gap> c5:=r1^3*r2^3*r1^-3*r2^-3;
( 1,20,10)( 4, 7,15)
gap> c6:=r2*r1^-1*r2^-1*r1;
( 1, 2,22)( 7, 8,17)
gap> c2*c6;
( 2,22,12)( 6, 8,17)
gap> rot90:=(2,13,12,22)(3,14,11,21)(5,16,9,19)(6,17,8,18)*r3^-1;
( 2,13,12,22)( 3,14,11,21)( 4,30,29,20,28,27,10,26,25,15,24,23)( 5,16, 9,19)
( 6,17, 8,18)
gap> c2*c6*rot90*c2*c6*rot90^-1;
( 2,12)( 6, 8)(13,22)(17,18)
gap> 
gap> 
gap> G1:=Group(r1,r2,r3,rot90);
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12), ( 1,13,14,15,16,17, 7,18,19,20,
 21,22), ( 4,23,24,15,25,26,10,27,28,20,29,30), ( 2,13,12,22)( 3,14,11,21)
( 4,30,29,20,28,27,10,26,25,15,24,23)( 5,16, 9,19)( 6,17, 8,18) )
gap> Size(G1);
21424936845312000
gap> G:=Group(r1,r2,r3);
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12), ( 1,13,14,15,16,17, 7,18,19,20,
 21,22), ( 4,23,24,15,25,26,10,27,28,20,29,30) )
gap> Size(G);
21424936845312000
gap> x1:=(2,13,12,22);
( 2,13,12,22)
gap> x1 in G;
false
gap> x2:=(2,1);
(1,2)
gap> x2 in G;
false
gap> x3:=(1,2,3);
(1,2,3)
gap> x3 in G;
false
gap> x4:=(1,2,12);
( 1, 2,12)
gap> x4 in G;
false
gap> N:=Size(G);
21424936845312000
gap> Factors(N);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
  3, 3, 3, 3, 3, 3, 5, 5, 5, 7, 7, 11, 13 ]
gap> Factors(Factorial(16));
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 7, 
  7, 11, 13 ]
gap> Factors(Factorial(16)*2^9);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 
  3, 3, 3, 3, 3, 5, 5, 5, 7, 7, 11, 13 ]
gap> Factors(Factorial(16)*2^10);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 
  3, 3, 3, 3, 3, 3, 5, 5, 5, 7, 7, 11, 13 ]
gap> #N=16!*2^{10}
gap> N/10^18;
653837184/30517578125
gap> N/10^17;
1307674368/6103515625
gap> N/10^16;
2615348736/1220703125
gap> N/10.0^16;
Syntax error: ; expected
N/10.0^16;
    ^
0
gap> #N is approx 2*10^{16}
gap> x5:=(1,7);
(1,7)
gap> x5 in G;
false
gap> x6:=(1,4,7);
(1,4,7)
gap> x6 in G;
false
gap> DG:=DerivedSubgroup(G);
Subgroup( Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12), ( 1,13,14,15,16,17, 7,
 18,19,20,21,22), ( 4,23,24,15,25,26,10,27,28,20,29,30) ), 
[ ( 1, 2,13)( 7, 8,18), ( 4, 5,23)(10,11,27), (15,16,25)(20,21,29), 
  ( 2, 3,13)( 8, 9,18), ( 2, 4,13)( 8,10,18), 
  ( 2, 4, 5,23,14,13, 3)( 8,10,11,27,19,18, 9)(16,17,25)(21,22,29), 
  ( 2,14,13)( 5,24,23)( 8,19,18)(11,28,27), 
  ( 1, 2, 5,24,14,13)( 3,23)( 7, 8,11,28,19,18)( 9,27)(16,17,26)(21,22,30), 
  ( 1,29,21,30,22)( 2, 4,24, 3,23,15, 5,13,14)( 7,25,16,26,17)( 8,10,28, 9,27,
     20,11,18,19), ( 2,13, 3, 6,24, 4,23, 5,14)( 8,18, 9,12,28,10,27,11,19)
    (16,17,26)(21,22,30) ] )
gap> Size(DG);
10712468422656000
gap> N1:=Size(DG);
10712468422656000
gap> N/N1;
2
gap> 
gap> PCore(G,2);
Subgroup( Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12), ( 1,13,14,15,16,17, 7,
 18,19,20,21,22), ( 4,23,24,15,25,26,10,27,28,20,29,30) ), 
[ (25,29)(26,30), (24,28)(26,30), (23,27)(26,30), (17,22)(26,30), 
  (16,21)(26,30), (15,20)(23,27)(25,29)(26,30), (14,19)(17,22), 
  (13,18)(16,21)(24,28)(25,29), ( 6,12)(23,27)(24,28)(25,29), ( 5,11)(25,29), 
  ( 4,10)(15,20)(23,27)(24,28)(25,29)(26,30), 
  ( 3, 9)( 5,11)( 6,12)(15,20)(23,27)(26,30), 
  ( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(24,28), 
  ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] )
gap> PG2:=PCore(G,2);
Subgroup( Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12), ( 1,13,14,15,16,17, 7,
 18,19,20,21,22), ( 4,23,24,15,25,26,10,27,28,20,29,30) ), 
[ (25,29)(26,30), (24,28)(26,30), (23,27)(26,30), (17,22)(26,30), 
  (16,21)(26,30), (15,20)(23,27)(25,29)(26,30), (14,19)(17,22), 
  (13,18)(16,21)(24,28)(25,29), ( 6,12)(23,27)(24,28)(25,29), ( 5,11)(25,29), 
  ( 4,10)(15,20)(23,27)(24,28)(25,29)(26,30), 
  ( 3, 9)( 5,11)( 6,12)(15,20)(23,27)(26,30), 
  ( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(24,28), 
  ( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12) ] )
gap> N2:=Size(PG2);
16384
gap> L2:=Factors(N2);
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
gap> Length(L2);
14
gap> N/2^14;
1307674368000
gap> Factorial(15);
1307674368000
gap> #G is S15 semidirect C2^14??
gap> 
gap> 
gap> Blocks(G,[1..30]);
[ [ 1, 7 ], [ 2, 8 ], [ 13, 18 ], [ 3, 9 ], [ 14, 19 ], [ 4, 10 ], 
  [ 15, 20 ], [ 5, 11 ], [ 23, 27 ], [ 16, 21 ], [ 25, 29 ], [ 6, 12 ], 
  [ 24, 28 ], [ 17, 22 ], [ 26, 30 ] ]
gap> B:=Blocks(G,[1..30]);
[ [ 1, 7 ], [ 2, 8 ], [ 13, 18 ], [ 3, 9 ], [ 14, 19 ], [ 4, 10 ], 
  [ 15, 20 ], [ 5, 11 ], [ 23, 27 ], [ 16, 21 ], [ 25, 29 ], [ 6, 12 ], 
  [ 24, 28 ], [ 17, 22 ], [ 26, 30 ] ]
gap> Length(B);
15
gap> #maybe G permutes these blocks but allows some swaps
gap> #within a block (not any swaps, else 2^15 rather than
gap> #2^14 would be the size of the normal subgroup)
gap> quit;