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) );; Size( cube ); Collected( Factors( last ) ); SizeScreen( [71, ] );; orbits := Orbits( cube, [1..48] ); cube1 := Operation( cube, orbits[1] ); Size( cube1 ); corners := Blocks( cube1, [1..24] ); cube1b := Operation( cube1, corners, OnSets ); Size( cube1b ); blockhom1 := OperationHomomorphism( cube1, cube1b );; Factors( Size( Kernel( blockhom1 ) ) ); IsElementaryAbelian( Kernel( blockhom1 ) ); cmpl1 := Stabilizer( cube1, [1,2,3,4,5,6,8,13], OnSets );; Size( cmpl1 ); Size( Intersection( cmpl1, Kernel( blockhom1 ) ) ); Closure( cmpl1, Kernel( blockhom1 ) ) = cube1; IsIsomorphism( OperationHomomorphism( cmpl1, cube1b ) ); (1,7,22) in cube1; (1,7,22)(2,20,14) in cube1; cube2 := Operation( cube, orbits[2] );; Size( cube2 ); edges := Blocks( cube2, [1..24] ); cube2b := Operation( cube2, edges, OnSets );; Size( cube2b ); blockhom2 := OperationHomomorphism( cube2, cube2b );; Factors( Size( Kernel( blockhom2 ) ) ); IsElementaryAbelian( Kernel( blockhom2 ) ); cmpl2 := Stabilizer(cube2,[1,2,3,4,5,6,7,9,10,12,14,16],OnSets);; IsIsomorphism( OperationHomomorphism( cmpl2, cube2b ) ); (1,11) in cube2; (1,11)(2,17) in cube2; Size( cube ); Size( cube1 ) * Size( cube2 ); (17,19)(11,8)(6,25) in cube; (7,28)(18,21) in cube; (17,19)(11,8)(6,25)(7,28)(18,21) in cube; cube.name := "cube";; Centre( cube ); quit;