[GAP Forum] order of conjugacy classes?

[Hongyi Zhao]

On Fri, Jun 17, 2022 at 10:00 PM R. K. Dennis <gap at rkd.math.cornell.edu> wrote:
>
> Dear Colleagues,
>
> I have a computation where the results are stored and displayed
> in a max by conj matrix.  Here max is the number of conjugacy
> classes of maximal subgroups and conj is the number of conjugacy
> classes of elements.
>
> The orders of the rows and columns corresponds to the classes.
>
> But gap doesn't generate the conjugacy classes of maximal subgroups
> in the same order every time; I don't know about classes of elements
> as I haven't noticed any difference in the matrix (but I suppose
> that might depend of the type of computations I'm doing).
>
> Ok, so my question is:
>
> Is there a way to get gap to present the results in the same order
> every time?  I suppose I could sort the results by order of subgroup
> size.  But that wouldn't necessarily uniquely order the representatives
> of the conjugacy classes of maximal subgroups.
>
> Is there anything reasonable to do?
>
> For conjugacy classes of elements, gap doesn't seem to sort by
> size or order, but does seem reasonably consistent, at least for
> the small examples I've looked at.
>
> Suggestions most welcome!


I hope the following example represented here [1] by Olexandr
Konovalov will give you some clues:

G:=Group((1,9,6,7,5)(2,10,3,8,4), (1,10,7,8)(2,9,4,6));
cc:=ConjugacyClasses(G);
Length(cc);
c:=cc[2];
for x in c do
  Print(x,"\n");
od;

# without constructing the whole list, and also use enumerator which
will give you a list-like behaviour:

enum:=Enumerator(c);
enum[2];
Position( enum, (3,7,9,4)(5,10,6,8) );
Position( enum, (3,7,6,8) );


[1] https://math.stackexchange.com/questions/2604094/list-conjugacy-classes-in-gap

>
> Thanks!
>
> Keith

Yours,
Hongsheng