To start GAP, type 'gap' on the command line.

For help, type '?' within GAP.  You can preface a command with a '?' to bring
up a help page about it.  For instance, '?Group' brings up the help page for
the 'Group' command.  Help pages will also give examples.  Try '?tutorial' if
you'd like a tutorial.  If the help page is a list of references, then '?1'
will will bring up the first one, etc.  Otherwise, '?<whatever they have written>'
will work.

To quickly type out commands, and to look for other commands, GAP offers tab
completion.  Just type out part of the word and press the TAB key.  GAP will
either complete the command out, complete it as much as possible before become
non-unique, or give you a list of possible completions if it cannot complete it
further.
gap> Double
    DoubleCentralizerOrbit
    DoubleCoset
    DoubleCosetRepsAndSizes
    DoubleCosets
    DoubleCosetsNC
    DoubleCosetsPcGroup
    DoubleHashArraySize
    DoubleHashDictSize
gap> Double



gap> 2+2;
4
gap> # cycles: acting on a set, multiplicaion, and conjugation
gap> 1^(1,2,3,4);
2
gap> (1,2,3,4) * (1,3,4,2);
(2,4,3)
gap> (1,2,3,4)^(1,3,4,2);
(1,4,2,3)
gap> # lists can be written in two ways
gap> [1,2,3,4,5,6];
[ 1, 2, 3, 4, 5, 6 ]
gap> [1..6];
[ 1 .. 6 ]
gap> # advanced list and conjugation use, conjugate each element
gap> List( [1,2], x -> x^(1,2,3,4) );
[ 2, 3 ]
gap> # permutation group (on [1..6]) & various properties
gap> G := Group( (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6));
Group([ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6) ])
gap> Order(G);
48
gap> Orbits(G, [1..11]);
[ [ 1, 5, 2, 4, 6, 3 ], [ 7 ], [ 8 ], [ 9 ], [ 10 ], [ 11 ] ]
gap> IsTransitive(G);
true
gap> IsPrimitive(G);
false
gap> RankAction(G);
3

gap> # verify the rank via the number of suborbits and orbitals
gap> S := Stabilizer(G, 1);
Group([ (3,6)(4,5), (3,4), (5,6) ])
gap> Orbits(S,[1..6]);     
[ [ 1 ], [ 2 ], [ 3, 5, 6, 4 ] ]
gap> Number(Orbits(S,[1..6]));
3
gap> Orbits(G, Tuples([1..6],2), OnTuples);
[ [ [ 1, 1 ], [ 2, 2 ], [ 5, 5 ], [ 6, 6 ], [ 4, 4 ], [ 3, 3 ] ], 
  [ [ 1, 2 ], [ 2, 1 ], [ 5, 6 ], [ 6, 5 ], [ 4, 3 ], [ 3, 4 ] ], 
  [ [ 1, 3 ], [ 2, 3 ], [ 1, 4 ], [ 2, 4 ], [ 5, 2 ], [ 6, 2 ], [ 5, 1 ], 
      [ 6, 1 ], [ 4, 6 ], [ 3, 6 ], [ 4, 5 ], [ 3, 5 ], [ 1, 6 ], [ 2, 6 ], 
      [ 1, 5 ], [ 2, 5 ], [ 4, 2 ], [ 3, 2 ], [ 4, 1 ], [ 3, 1 ], [ 5, 3 ], 
      [ 6, 3 ], [ 5, 4 ], [ 6, 4 ] ] ]
gap> Number(Orbits(G, Tuples([1..6],2), OnTuples));
3

gap> Random(G);
(1,4,2,3)(5,6)
gap> Random(G);
(1,2)(3,6,4,5)
gap> Random(G);
(1,2)(5,6)
gap> # consider Sym(6) acting on the cosets of G as a group
gap> Index( SymmetricGroup(6), G);
15
gap> perms := CosetTableBySubgroup( SymmetricGroup(6), G );
[ [ 2, 1, 4, 6, 3, 10, 9, 5, 12, 8, 13, 7, 14, 11, 15 ],
  [ 2, 1, 5, 3, 8, 4, 12, 10, 7, 6, 14, 9, 11, 13, 15 ],
  [ 1, 3, 2, 7, 9, 11, 4, 13, 5, 10, 6, 12, 8, 15, 14 ],
  [ 1, 3, 2, 7, 9, 11, 4, 13, 5, 10, 6, 12, 8, 15, 14 ] ]
gap> List(perms, PermList);
[ (1,2)(3,4,6,10,8,5)(7,9,12)(11,13,14), (1,2)(3,5,8,10,6,4)(7,12,9)(11,14,13),
  (2,3)(4,7)(5,9)(6,11)(8,13)(14,15), (2,3)(4,7)(5,9)(6,11)(8,13)(14,15) ]
gap> Order(Group(List(perms, PermList)));
720

gap> DoubleCosets(SymmetricGroup(6),G,G);
[ DoubleCoset(Group( [ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6)
     ] ),(),Group( [ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6) ] )),
  DoubleCoset(Group( [ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6) ] ),
    ( 4, 5),Group( [ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6) ] )),
  DoubleCoset(Group( [ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6) ] ),
    ( 2, 3)( 4, 5),Group( [ (1,2), (3,4), (5,6), (1,3,2,4), (1,5,2,6), (3,5,4,6)
     ] )) ]

gap> # programming example
gap> for elts in G do
> Print( elts , Order(elts), "\n");
> od;
()1
(5,6)2
(3,4)2
(3,4)(5,6)2
(3,5)(4,6)2
(3,5,4,6)4
(3,6,4,5)4
(3,6)(4,5)2
(1,2)2
(1,2)(5,6)2
(1,2)(3,4)2
(1,2)(3,4)(5,6)2
(1,2)(3,5)(4,6)2
(1,2)(3,5,4,6)4
(1,2)(3,6,4,5)4
(1,2)(3,6)(4,5)2
(1,3)(2,4)2
(1,3)(2,4)(5,6)2
(1,3,2,4)4
(1,3,2,4)(5,6)4
(1,3,5)(2,4,6)3
(1,3,5,2,4,6)6
(1,3,6,2,4,5)6
(1,3,6)(2,4,5)3
(1,4,2,3)4
(1,4,2,3)(5,6)4
(1,4)(2,3)2
(1,4)(2,3)(5,6)2
(1,4,6,2,3,5)6
(1,4,6)(2,3,5)3
(1,4,5)(2,3,6)3
(1,4,5,2,3,6)6
(1,5,3)(2,6,4)3
(1,5,4,2,6,3)6
(1,5,3,2,6,4)6
(1,5,4)(2,6,3)3
(1,5)(2,6)2
(1,5,2,6)4
(1,5)(2,6)(3,4)2
(1,5,2,6)(3,4)4
(1,6,4,2,5,3)6
(1,6,3)(2,5,4)3
(1,6,4)(2,5,3)3
(1,6,3,2,5,4)6
(1,6,2,5)4
(1,6)(2,5)2
(1,6,2,5)(3,4)4
(1,6)(2,5)(3,4)2
gap> for elts in G do
> Print( elts ,"\t", Order(elts), "\n");
> od;
()      1
(5,6)   2
(3,4)   2
(3,4)(5,6)      2
(3,5)(4,6)      2
(3,5,4,6)       4
(3,6,4,5)       4
(3,6)(4,5)      2
(1,2)   2
(1,2)(5,6)      2
(1,2)(3,4)      2
(1,2)(3,4)(5,6) 2
(1,2)(3,5)(4,6) 2
(1,2)(3,5,4,6)  4
(1,2)(3,6,4,5)  4
(1,2)(3,6)(4,5) 2
(1,3)(2,4)      2
(1,3)(2,4)(5,6) 2
(1,3,2,4)       4
(1,3,2,4)(5,6)  4
(1,3,5)(2,4,6)  3
(1,3,5,2,4,6)   6
(1,3,6,2,4,5)   6
(1,3,6)(2,4,5)  3
(1,4,2,3)       4
(1,4,2,3)(5,6)  4
(1,4)(2,3)      2
(1,4)(2,3)(5,6) 2
(1,4,6,2,3,5)   6
(1,4,6)(2,3,5)  3
(1,4,5)(2,3,6)  3
(1,4,5,2,3,6)   6
(1,5,3)(2,6,4)  3
(1,5,4,2,6,3)   6
(1,5,3,2,6,4)   6
(1,5,4)(2,6,3)  3
(1,5)(2,6)      2
(1,5,2,6)       4
(1,5)(2,6)(3,4) 2
(1,5,2,6)(3,4)  4
(1,6,4,2,5,3)   6
(1,6,3)(2,5,4)  3
(1,6,4)(2,5,3)  3
(1,6,3,2,5,4)   6
(1,6,2,5)       4
(1,6)(2,5)      2
(1,6,2,5)(3,4)  4
(1,6)(2,5)(3,4) 2
gap> Random(G);
(1,4)(2,3)(5,6)
gap> Random(G);
(1,6,4)(2,5,3)
gap>
gap> # Example for finding empirically fining how many permutations are fpf
gap>
gap> Read("gap/FixedPointFree.gap");
gap> IsFixedPointFree( [1..4], (1,2)(3,4) );
true
gap> IsFixedPointFree( [1..4], (1,2)(3,5) );
false
gap> IsFixedPointFree( [1..4], (1,2,3) );
false
gap> for N in [1..12] do
> SymmetricFPF_Percent(N);
> od;
Sym(1): 0 in 1000 randomly selected.
Sym(2): 525 in 1000 randomly selected.
Sym(3): 331 in 1000 randomly selected.
Sym(4): 335 in 1000 randomly selected.
Sym(5): 369 in 1000 randomly selected.
Sym(6): 360 in 1000 randomly selected.
Sym(7): 384 in 1000 randomly selected.
Sym(8): 363 in 1000 randomly selected.
Sym(9): 359 in 1000 randomly selected.
Sym(10): 353 in 1000 randomly selected.
Sym(11): 371 in 1000 randomly selected.
Sym(12): 330 in 1000 randomly selected.
gap> SymmetricFPF_Percent(20);
Sym(20): 353 in 1000 randomly selected.
gap> SymmetricFPF_Percent(20);
Sym(20): 372 in 1000 randomly selected.
gap> SymmetricFPF_Percent(100);
Sym(100): 390 in 1000 randomly selected.