[GAP Forum] SmallGroup(64,177)

[Joe Bohanon]

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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