###################################################################
#David Joyner, 11-2006
#
#
#
#                         +--------------+
#                         |  1    2    3 |
#                         |  4  top    5 |
#                         |  6    7    8 |
#          +--------------+--------------+--------------+--------------+
#          |  9   10   11 | 17   18   19 | 25   26   27 | 33   34   35 |
#          | 12  left  13 | 20 front  21 | 28 right  29 | 36  rear  37 |
#          | 14   15   16 | 22   23   24 | 30   31   32 | 38   39   40 |
#          +--------------+--------------+--------------+--------------+
#                         | 41   42   43 |
#                         | 44 bottom 45 |
#                         | 46   47   48 |
#                         +--------------+
#
###################################################################


U:=( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19);
L:=( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35);
F:= (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11);
R:=(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24);
B:=(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27);
D:=(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40);


cube := Group(U,L,F,R,B,D);;
#gap> Size(cube);
#43252003274489856000

square_group := Group(U^2,L^2,F^2,R^2,B^2,D^2);;
#gap> Size(square_group);
#663552
#gap> IsAbelian(square_group);
#false

square_two_faces := Group(U^2,R^2);;
#gap> Size(square_two_faces);
#12
#gap> IsAbelian(square_two_faces);
#false

two_faces:= Group(U,R);  ## 2-faced cube
#gap> Size(two_faces);
#73483200


#LoadPackage("grape");
#Gamma := CayleyGraph( two_faces, [U,U^(-1), R, R^(-1)] );;
#Diameter( Gamma );
#Gammatoo big?

N1:= Size( cube );
Collected( Factors(N1) );

N2:=Size(square_group);
Collected( Factors(N2) );

N3:=Size(two_faces);
Collected( Factors(N3) );

Kgens:=[];
for g in G do 
 if (8^g in [8,19,25]) and (3^g in [3,27,33]) then 
  Kgens:=Concatenation([g],Kgens);
 fi; 
od;
K:=Group(Kgens);
#<permutation group with 1620 generators>
#gap> Size(K);
#1620

Kgens:=[];
for g in G do 
 if (8^g in [8,19,25]) and (3^g in [3,27,33]) and (6^g in [6,11,17]) then 
  Kgens:=Concatenation([g],Kgens);
 fi; 
od;
K:=Group(Kgens);
#<permutation group with 1620 generators>
#gap> IsAbelian(K);
#true
#gap> IsNormal(two_faces,K);
#true
#gap> C := Complementclasses(two_faces,K);;
#gap> Length(C);
#3
#gap> IdGroup(C[1]); IdGroup(H);
#[ 120, 34 ]
#[ 120, 34 ]


Kgens:=[];
for g in two_faces do 
 if (8^g in [8,19,25]) and (3^g in [3,27,33]) and (6^g in [6,11,17])\
    and (1^g in [1,9,35]) and (24^g in [24,30,43]) and (32^g in [32,38,48])\
    and (7^g = 7) and (28^g = 28) then 
  Kgens:=Concatenation([g],Kgens);
 fi; 
od;
K:=Group(Kgens);
#<permutation group with 14580 generators>
#gap> IsAbelian(K);
#true
#gap> IsNormal(two_faces,K);
#false

Kgens:=[];
for g in two_faces do 
 if (8^g in [8,19,25]) and (3^g in [3,27,33]) and (6^g in [6,11,17])\
    and (1^g in [1,9,35]) and (24^g in [24,30,43]) and (32^g in [32,38,48]) then 
  Kgens:=Concatenation([g],Kgens);
 fi; 
od;
K:=Group(Kgens);
#<permutation group with 612360 generators>
#gap> IsAbelian(K);
#false
IsNormal(two_faces,K);
#true