Janko Group and subgroups.

Question

I want to work with the Janko simple group $J_2$ using the computer, and if it is possible, the Janko simple group $J_4$ too. In specific, I want to take certain subgroups of it and compute their lattice subgroup. Other question is if exists accesible bibliography about those groups to read about its subgroups and others characteristics of them.

I was working in GAP, but it doesn't support the $J_2$ group, because after certain point using the commands, it breaks and doesn't do correctly the commands based on it. I'm trying to use Magma, but it doesn't have a command to obtain the subgroups of a certain group, only conjugation classes, and brings me different solutions against the (mathematically proved) solutions that GAP gives. Any help will be really appreciated.

Answer 1

In Magma

> S := [ s`subgroup : s in Subgroups(G) ];
> SS:= &cat[[s^t : t in Transversal(G,Normaliser(G,s))] : s in S];        
> #SS;
1104344

gives a list of all 1104344 subgroups of $J_2$. I am sure you can do it in a similar way in GAP by starting with $\mathtt{ConjugacyClassesSubgroups}(G)$.

I fin it unlikely that you will succeed in computing a list of all subgroups of $J_4$.

Answer 2

In GAP (at least in principle -- my laptop did not have enough memory to store all of J2):

g:=SimpleGroup("J2");
u:=ConjugacyClassesSubgroups(g);;
v:=Concatenation(List(u,Elements));;

You will get repeatedly (after exceeding 2GB, 4GB, etc of memory) a warning from GAP "Error, reached the pre-set memory limit". This is a safety feature to stop a computing task potentially bringing the whole computer to its knees. Simply enter

return;

to continue the calculation.