> <subgroup lattice of Sym( [ 1 .. 3 ] ), 4 classes, 6 subgroups>
> I don't understand how I can construct the full lattice when I know only
> the number of Conjugacy Classes and the number of Subgroups.
Once you computed the subgroup lattice, other commands help you to find
out more about it. An Example is "MaximalSubgroups". Let's see how this
works on your example:
gap> g:=SymmetricGroup(3);; gap> lat:=LatticeSubgroups(g); <subgroup lattice of Sym( [ 1 .. 3 ] ), 4 classes, 6 subgroups> gap> classes:=ConjugacyClassesSubgroups(lat); [ Group( () )^G, Group( [ (2,3) ] )^G, Group( [ (1,2,3) ] )^G, SymmetricGroup( [ 1 .. 3 ] )^G ] gap> lengths:=List(classes, Size); [1, 3, 1, 1 ]
# Hence the lattice has 4 conjugacy classes, one of length 3 and 3 of
# length 1. These are three normal subgroups!
# Now we take representatives in the classes...
gap> repr:=List(classes, Representative); [ Group(()), Group([ (2,3) ]), Group([ (1,2,3) ]), Sym( [ 1 .. 3 ] ) ] gap> sizes:=List(repr, Size); [ 1, 2, 3, 6 ]
# OK. Now we know the groups in our classes.
gap> maxsg:=MaximalSubgroupsLattice(lat); [ [ ], [ [ 1, 1 ] ], [ [ 1, 1 ] ], [ [ 3, 1 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ] ] ]
# Here we see that the groups in the first class have no subgroups. Of
# course! The groups in that class are trivial.
# A representative of the second class contains a group of the first class
# as a maximal subgroup. The same is true for a representative of the
# third class. A representative of the fourth class (i.e. the group Sym(3)
# itself) contains the group of the third class and all three groups of
# the second class as maximal subgroups.
With this information you can make a picture of the subgroup lattice of
Sym(3). More commands like this are available in GAP: just check the
manual. As an exercise, you should try the command
I have some experience with subgroup lattices in GAP. If you need more
information, I'll be happy to help you. Some years ago, I wrote a program
which generates a LaTeX file with the subgroup lattice of a given group
in the form of a table. This enables you to work in the subgroup lattice
if needed. If you really want a picture, you should take a
look at xgap but beware that subgroup lattices rapidly become very
complicated and difficult to draw.