peter@eiger:~$ gap
 ┌───────┐   GAP 4.12.1 of 2022-10-20
 │  GAP  │   https://www.gap-system.org
 └───────┘   Architecture: x86_64-pc-linux-gnu-default64-kv8
 Configuration:  gmp 6.3.0, GASMAN, readline
 Loading the library and packages ...
 Packages:   Alnuth 3.2.1, AtlasRep 2.1.6, AutPGrp 1.11, CTblLib 1.3.6, 
             FactInt 1.6.3, FGA 1.4.0, GAPDoc 1.6.6, IO 4.8.1, LAGUNA 3.9.5, 
             Polycyclic 2.16, PrimGrp 3.4.4, SmallGrp 1.5.3, TomLib 1.2.9, 
             TransGrp 3.6.4, utils 0.82
 Try '??help' for help. See also '?copyright', '?cite' and '?authors'
gap> (1,2)*(1,3);
(1,2,3)
gap> cube := Group(
> ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),
> ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),
> (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),
> (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
> (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),
> (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) );
<permutation group with 6 generators>
gap> Size( cube );
43252003274489856000
gap> time;
13
gap> Factorial(48);
12413915592536072670862289047373375038521486354677760000000000
gap> (6,17,11) in cube;
false
gap> orbits := Orbits( cube);
[ [ 1, 3, 17, 14, 8, 38, 9, 41, 19, 48, 22, 6, 30, 33, 43, 11, 46, 40, 24, 
      27, 25, 35, 16, 32 ], 
  [ 2, 5, 12, 7, 36, 10, 47, 4, 28, 45, 34, 13, 29, 44, 20, 42, 26, 21, 37, 
      15, 31, 18, 23, 39 ] ]
gap> cube1 := Action( cube, orbits[1] );
<permutation group with 6 generators>
gap> Size( cube1 );
88179840
gap> cube2 := Action( cube, orbits[2] );
<permutation group with 6 generators>
gap> Size( cube2 );
980995276800
gap> Size( cube1 )* Size( cube2 );
86504006548979712000
gap> F := FreeGroup( "u", "l", "f", "r", "b", "d" );
<free group on the generators [ u, l, f, r, b, d ]>
gap> hom := GroupHomomorphismByImages( F, cube, GeneratorsOfGroup(F), GeneratorsOfGroup(cube) );
[ u, l, f, r, b, d ] -> 
[ (1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)(11,35,27,19), 
  (1,17,41,40)(4,20,44,37)(6,22,46,35)(9,11,16,14)(10,13,15,12), 
  (6,25,43,16)(7,28,42,13)(8,30,41,11)(17,19,24,22)(18,21,23,20), 
  (3,38,43,19)(5,36,45,21)(8,33,48,24)(25,27,32,30)(26,29,31,28), 
  (1,14,48,27)(2,12,47,29)(3,9,46,32)(33,35,40,38)(34,37,39,36), 
  (14,22,30,38)(15,23,31,39)(16,24,32,40)(41,43,48,46)(42,45,47,44) ]
gap> r := Random( cube );
(1,41,40,8,6,35,22,46,19,11,9,16,14,25,17)(2,39,28,5,13,29,12,18,42,45)(3,43,
32,33,24,38,27,30,48)(4,15)(7,23,31,34,47,21,26,20,36,37)(10,44)
gap> pre := PreImagesRepresentative( hom, r );
u*r*b^-1*u^-1*r*b*d*b^-1*f*u^2*d^-1*b^-1*d*f^-1*u*l*f^-1*r*u*r^-1*u^2*d*f*d^-1\
*l*(l*u^-1)^2*u^-1*l^-1*u^-1*l*u^-1*b*l*b^-1*l^-1*u*f*u*r*u^-1*r^-1*f^-1*u^-2*\
l^-1*u^-1*l*u^2*f*r*u*r^-1*u^-1*f^-1*u*l^-1*u^-1*b*l^-1*b^-1*l*u*(l*u^-1)^2*l^\
-1*u*f^-1*u^-1*l*u*l^-1*f^2*u*f^-1*l*u*l^-1*f*l*u^-1*l^-1*u^-1*f^-1*u^-2*l*f*u\
*f^-1*u^-1*l^-1*f*r*u^-1*r^-1*f^-1*l^-1*u*l
gap> quit;