why can not define G := Group((1),(3,2));;

Question

gap> G := Group((1),(3,2));;
Error, usage: Group(<gen>,...), Group(<gens>), Group(<gens>,<id>) called from
<function "Group">( <arguments> )
 called from read-eval loop at line 6 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue

before finding character table of permutation group S3 - 1,3,2

got error in above command

Answer 1

Now after the question is no longer on hold, I can reply with more details:

1) The error about the wrong usage of Group usually means that the group can not be generated by the arguments. For example,

gap> Group(1);
#I  no groups of cyclotomics allowed because of incompatible ^
Error, usage: Group(<gen>,...), Group(<gens>), Group(<gens>,<id>) called from
<function "Group">( <arguments> )

2) What's then wrong with the arguments in the question? See how GAP evaluates various inputs:

gap> (3,2); # works OK
(2,3)
gap> ();    # this is the notation for the identity permutation
()
gap> (1);   # this is an integer one in brackets
1

Thus, Group((1),(3,2)) tries to generate a group with an integer 1 and a permutation (3,2) as generators, what obviously does not make sense

3) What does (1) in the original question actually mean? If this has to be an identity permutation, one should use () - however, there is no need to add the identity element of the group to the list of generators, so Group((3,2)) just suffices:

gap> Group((3,2));
Group([ (2,3) ])

4) This may be not the intended group, however, since the question contains "permutation group S3 - 1,3,2" - it's unclear what is the meaning of "1,3,2" here, but if the intention was to create a symmetric group of permutations of degree 3, here there are several ways to achieve this:

gap> Group((1,2),(1,2,3));
Group([ (1,2), (1,2,3) ])
gap> SymmetricGroup(3);
Sym( [ 1 .. 3 ] )

5) Finally, the following hint about character tables may be useful:

One can compute the character table "on-fly" for a given group, and if some methods depend on random states, the result may differ each time you call CharacterTable (conjugacy classes may be ordered in a different way). For groups whose character tables are available from The GAP Character Table Library, the table will be retrieved from the library so the result will be the same each time. Compare Display(CharacterTable(SymmetricGroup(3))); and Display(CharacterTable("S3")); to see the difference.