gap> Read("FactorizationProgram.gap");



gap> G:=SL(2,8);;
gap> Factors(Order(G));
[ 2, 2, 2, 3, 3, 7 ]
gap> a:=12;;
gap> Factors(a);
[ 2, 2, 3 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 3, 7 ]

gap> H:=Group([[[0*Z(2), Z(2)^0],[Z(2)^0,Z(2)^0]],[[0*Z(2), Z(2)^0],[Z(2)^0, 0*Z(2)]]]);;
gap> Factors(Order(H));
[ 2, 3 ]
gap> K:=SylowSubgroup(G,7);;
gap> Factors(Order(K));
[ 7 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> a:=21;;
gap> Factors(a);
[ 3, 7 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 2, 3 ]

gap> H:=SylowSubgroup(G,7);;
gap> Factors(Order(H));
[ 7 ]
gap> K:=SylowSubgroup(G,2);;
gap> Factors(Order(K));
[ 2, 2, 2 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true



gap> G:=PSL(2,13);;
gap> Factors(Order(G));
[ 2, 2, 3, 7, 13 ]
gap> a:=21;;
gap> Factors(a);
[ 3, 7 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 13 ]

gap> H:=SylowSubgroup(G,7);;
gap> Factors(Order(H));
[ 7 ]
gap> K:=MaximalSubgroupClassReps(MaximalSubgroupClassReps(G)[1])[2];;
gap> Factors(Order(K));
[ 2, 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true



gap> G:=PSL(2,17);;
gap> Factors(Order(G));
[ 2, 2, 2, 2, 3, 3, 17 ]
gap> a:=48;;
gap> Factors(a);
[ 2, 2, 2, 2, 3 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 3, 17 ]

gap> H:=SylowSubgroup(G,2);;
gap> Factors(Order(H));
[ 2, 2, 2, 2 ]
gap> K:=SylowSubgroup(G,17);;
gap> Factors(Order(K));
[ 17 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true



gap> G:=PSL(2,19);;
gap> Factors(Order(G));
[ 2, 2, 3, 3, 5, 19 ]
gap> a:=36;;
gap> Factors(a);
[ 2, 2, 3, 3 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 5, 19 ]

gap> H:=MaximalSubgroupClassReps(G)[5];;
gap> Factors(Order(H));
[ 2, 3, 3 ]
gap> K:=SylowSubgroup(G,19);;
gap> Factors(Order(K));
[ 19 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> a:=38;;
gap> Factors(a);
[ 2, 19 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 3, 3, 5 ]

gap> H:=SylowSubgroup(G,19);;
gap> Factors(Order(H));
[ 19 ]
gap> K:=MaximalSubgroupClassReps(G)[5];;
gap> Factors(Order(K));
[ 2, 3, 3 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> a:=45;;
gap> Factors(a);
[ 3, 3, 5 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 19 ]

gap> H:=SylowSubgroup(G,3);;
gap> Factors(Order(H));
[ 3, 3 ]
gap> K:=SylowSubgroup(G,19);;
gap> Factors(Order(K));
[ 19 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true



gap> G:=PSL(2,23);;
gap> Factors(Order(G));
[ 2, 2, 2, 3, 11, 23 ]
gap> a:=44;;
gap> Factors(a);
[ 2, 2, 11 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 3, 23 ]

gap> H:=MaximalSubgroupClassReps(G)[5];;
gap> Factors(Order(H));
[ 2, 11 ]
gap> K:=SylowSubgroup(G,23);;
gap> Factors(Order(K));
[ 23 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> a:=46;;
gap> Factors(a);
[ 2, 23 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 3, 11 ]

gap> H:=SylowSubgroup(G,23);;
gap> Factors(Order(H));
[ 23 ]
gap> K:=MaximalSubgroupClassReps(G)[5];;
gap> Factors(Order(K));
[ 2, 11 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> a:=66;;
gap> Factors(a);
[ 2, 3, 11 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 23 ]

gap> H:=MaximalSubgroupClassReps(G)[5];;
gap> Factors(Order(H));
[ 2, 11 ]
gap> K:=SylowSubgroup(G,23);;
gap> Factors(Order(K));
[ 23 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> a:=69;;
gap> Factors(a);
[ 3, 23 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 2, 11 ]

gap> H:=SylowSubgroup(G,23);;
gap> Factors(Order(H));
[ 23 ]
gap> K:=MaximalSubgroupClassReps(G)[5];;
gap> Factors(Order(K));
[ 2, 11 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true


gap> G:=PSL(2,25);;
gap> Factors(Order(G));
[ 2, 2, 2, 3, 5, 5, 13 ]
gap> a:=39;;
gap> Factors(a);
[ 3, 13 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 2, 5, 5 ]

gap> H:=SylowSubgroup(G,13);;
gap> Factors(Order(H));
[ 13 ]
gap> K:=MaximalSubgroupClassReps(MaximalSubgroupClassReps(PSL(2,25))[1])[2];;
gap> Factors(Order(K));
[ 2, 2, 5, 5 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> G:=PSL(2,27);;
gap> Factors(Order(G));
[ 2, 2, 3, 3, 3, 7, 13 ]
gap> a:=18;
gap> Factors(a);
[ 2, 3, 3 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 3, 7, 13 ]

gap> H:=MaximalSubgroupClassReps(SylowSubgroup(G,3))[1];;
gap> Factors(Order(H));
[ 3, 3 ]
gap> K:=MaximalSubgroupClassReps(G)[3];;
gap> Factors(Order(K));
[ 2, 2, 3, 7, 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=21;;
gap> Factors(a);
[ 3, 7 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 3, 3, 13 ]

gap> H:=SylowSubgroup(G,7);;
gap> Factors(Order(H));
[ 7 ]
gap> K:=MaximalSubgroupClassReps(G)[3];;
gap> Factors(Order(K));
[ 2, 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=36;;
gap> Factors(a);
[ 2, 2, 3, 3 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 3, 7, 13 ]

gap> H:=MaximalSubgroupClassReps(G)[4];;
gap> Factors(Order(H));
[ 2, 2, 3 ]
gap> K:=SylowSubgroup(G,13);;
gap> Factors(Order(K));
[ 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=42;;
gap> Factors(a);
[ 2, 3, 7 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 3, 3, 13 ]

gap> H:=MaximalSubgroupClassReps(MaximalSubgroupClassReps(G)[2])[1];;
gap> Factors(Order(H));
[ 2, 7 ]
gap> K:=SylowSubgroup(G,13);;
gap> Factors(Order(K));
[ 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=52;;
gap> Factors(a);
[ 2, 2, 13 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 3, 3, 3, 7 ]

gap> H:=MaximalSubgroupClassReps(G)[3];;
gap> Factors(Order(H));
[ 2, 13 ]
gap> K:=SylowSubgroup(G,13);;
gap> Factors(Order(K));
[ 3, 3, 3 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=54;;
gap> Factors(a);
[ 2, 3, 3, 3 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 7, 13 ]

gap> H:=SylowSubgroup(G,3);;
gap> Factors(Order(H));
[ 3, 3, 3 ]
gap> K:=MaximalSubgroupClassReps(G)[3];;
gap> Factors(Order(K));
[ 2, 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=63;;
gap> Factors(a);
[ 3, 3, 7 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 3, 13 ]

gap> H:=MaximalSubgroupClassReps(SylowSubgroup(G,3))[1];;
gap> Factors(Order(H));
[ 3, 3 ]
gap> K:=MaximalSubgroupClassReps(G)[3];;
gap> Factors(Order(K));
[ 2, 13 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=78;;
gap> Factors(a);
[ 2, 3, 13 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 3, 3, 7 ]

gap> H:=MaximalSubgroupClassReps(G)[3];;
gap> Factors(Order(H));
[ 2, 13 ]
gap> K:=MaximalSubgroupClassReps(SylowSubgroup(G,3))[1];;
gap> Factors(Order(K));
[ 3, 3 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> a:=91;;
gap> Factors(a);
[ 7, 13 ]
gap> b:=Order(G)/a;;
gap> Factors(b);
[ 2, 2, 3, 3, 3 ]

gap> H:=SylowSubgroup(G,13);;
gap> Factors(Order(H));
[ 13 ]
gap> K:=SylowSubgroup(G,3);;
gap> Factors(Order(K));
[ 3, 3, 3 ]

gap> R:=FactorizationIntoSubsets(G,H,K,a);;
Matrix of double cosets was built.
gap> A:=R[1];;
gap> Size(A)=a;
true
gap> B:=R[2];;
gap> Size(B)=b;
true
gap> G=ProdSets(A,B);
true

gap> quit;