If you don't specifically need the group in this exact form, it also
works as Z8 : D8.
If you plan to be doing a lot of things like this, you should check out
the link below that I created a few years ago.
http://www.joebohanon.com/math/64grps.txt
Replace 64 with 32 to get those groups. You should double check my work
by copying-and-pasting the code from the declaration of "f" down to the
declaration of "g" then running:
IdGroup(g)
and making sure it returns [64,177] (or whatever number you need). I
created that page for exactly the reason you described. My dissertation
worked a lot with generators and relations and it was frustrating to try
to deduce them from the internal structure of the small groups as
PC-groups rather than some more natural finite presentation that lines
up well with StructureDescription.
Now, all that to say this. While the finite presentations I list are
good for, say, describing the group in a paper, almost any computation
you'd want to do in GAP works much faster with the PC-presentation. So
what you ought to do is
G:=SmallGroup(64,177);
iso:=IsomorphismGroups(g,G);
Then use the isomorphism to move back-and-forth when you need to. Or if
the IsomorphismGroups command stalls, do IsomorphismPcGroup first then
IsomorphicGroups with the image. For this specific group, when you run
"Center" on g you get a group with 33 generators, in spite of the center
having size 4.
I've thought about doing this for groups of order 128, but I did 64
almost entirely by hand and there are 10 times the groups of order 128.
A fairly difficult thing to do is to find a decent way to present the
group. StructureDescription doesn't do anything with central products
(as far as I know). Look at what the output gives you on
SmallGroup(32,50) which is Q8 * D8. All that to say that for any given
group, there are many way to present it.
Joe
Dan Lanke wrote:
Dear GAP Forum,
Gap tells me that the structure description of SmallGroup(64,177) is
(C2 x D16) : C2. How do I determine explicitly the action of C2 on (C2 x D16), so that I can do some computations by hand?
Thanks,
D.
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