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Dear GAP-Forum,

Katharina Geissler wrote: (translation by me)

I'm searching a function for the transitive groups of degree 12 to display

how the several subgroups of S12 are contained within each other.

What you're asking for are actually two things. The second is a graphical

display. In principle you could persuade 'GraphicLattice' to display a

partial lattice for the transitive groups. See the ongoing discussion about

XGAP and its possible extensions. We are currently discussing several

possibilities, but so far nothing specific has been planned.

Nevertheless I doubt that this is what you really want, unless you have a

BIG monitor: there are 301 transitive groups of degree 12. Displaying all

containments of conjugates will probably result in a mess of lines on the

screen which is not really usable. Anyhow, if you're using pen and paper you

could still try to produce such a picture based on the information you would

provide to 'GraphicLattice'. (BTW.: If you finally get such a picture I

would be VERY interested to get a copy of it. So far I've been too lazy

to do it myself.)

The first (and bigger) problem is to get the actual containment information.

So far there is no function to compute this with a single command in GAP.

I will describe, however, how you can compute it yourself, but it might

take you some time to do so:

The process will yield not only information *whether* a group is contained

in another, but also information how many conjugacy classes exist if the

subgroup is maximal. (You will need this information if you want to identify

Galois groups, as I suppose. You should note as well, that GAP already

contains information about resolvents distinguishing the groups. Probably

this is of help. Write to me directly if you want more information about this.)

If you can compute representatives of the conjugacy classes of maximal

subgroups of each transitive group, you are done. Non-maximal containment

simply follows by induction. 265 of the transitive groups are solvable. By

converting them to an AgGroup and then to a SpecialAgGroup, you can compute

representatives of the conjugacy classes of maximal subgroups and

transfer them (using the components .bijection in the SAgGroup and the

AgGroup) back in the permutation group. There, you select the ones which are

transitive. The command 'TransitiveIdentification' then tells you for each

of the representatives the number in the list of transitive groups, avoiding

conjugacy tests in S12.

This leaves 36 non-solvable groups. Coping with S12 and A12 is quite simple,

as the maximal subgroups are classified already (the imprimitive ones are

wreath products, the primitive ones are dealt with in the ATLAS).

8 of the remaining groups are wreath products. I have procedures to get

representatives of the conjugacy classes of transitive subgroups for these

groups, that I can provide to you if you want. However, it is not hard to

classify their maximal subgroups.

Of the remaining 26 groups, 18 are of size smaller 10000. You can use the

subgroup lattice program to get their maximal subgroups.

Most of the other groups are normal of small index in a wreath product.

Using this information one can describe the transitive subgroups of them.

(Again, I have functions to deal with this case. Write to me if you want

further information.)

Remaining is M12, whose maximal subgroups are given in the ATLAS.

In all these cases, after getting the maximal subgroups, treatment is the

same as in the solvable case.

Going through this process is a tour de force. However I can't imagine an

easier way (except persuading someone else to do it, but that's exactly

what I'm doing here).

I hope this helps. If anything in my description is unclear please ask.

Alexander Hulpke

-- Lehrstuhl D fuer Mathematik, RWTH, Templergraben 64, 52056 Aachen, Germany,

eMail: Alexander.Hulpke@math.rwth-aachen.de

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