Dear Forum,

Further to the message I have just sent, regarding problems with the
software 'miles' which runs the GAP Forum, it has come to my attention that
the following mail may not have been successfully delivered to all
subscribers to the forum. If you have already seen this, then I am sorry
for the duplication.

John.

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Forwarded message:

Date: Fri, 2 Aug 2002 20:00:16 -0400
From: Igor Schein <igor@txc.com>
To: GAP Forum <GAP-Forum@dcs.st-and.ac.uk>
Subject: Re: Galois group question
Reply-To: igor@txc.com

Dear Alexander,

On Fri, Aug 02, 2002 at 10:43:13PM +0300, Alexander B. Konovalov wrote:
> Dear Igor, Dear Forum,
>
> Really, the quaternion group of order 16 is just SmallGroup(16,9) and
> the one of order 32 - just SmallGroup(32,20).
>
> But indicate please what do you mean by QD16? The reason is that
> SmallGroup(16,8) and SmallGroup(32,19) are being semi-dihedral groups,
> but not quasi-dihedral ones.

Well, here's what I was refering to

\\ GAP4
gap> Gap3CatalogueIdGroup(SmallGroup(16,8));
[ 16, 13 ]

\\ GAP3
gap> GroupId(SolvableGroup(16,13));
rec(
   catalogue := [ 16, 13 ],
   size := 16,
   names := [ "QD16" ],
   pGroupId := 8 )

So semidihedral group is denoted as QD16, which I understand to be
quasidihedral. Why terminology discrepancy?

Below is a small function which returns IdGroup for 2-groups with the
cyclic subgroup of index 2, namely, the dihedral, quaternion,
semidihedral and quasidihedral groups, which confrim this.

groups:=function(n)
local F,a,b,D,Q,SD,QD;
F:=FreeGroup("a","b");
a:=F.1; b:=F.2;
D :=F/[a^(n/2), b^2,         b^-1*a*b*a];
Q :=F/[a^(n/2), b^2*a^(n/4), b^-1*a*b*a];
SD:=F/[a^(n/2), b^2,         b^-1*a*b*a^(1-(n/4))];
QD:=F/[a^(n/2), b^2,         b^-1*a*b*a^(-(n/4)-1)];
return List([D,Q,SD,QD],IdGroup);
end;

Thanks a lot for the code.

Igor

End of forwarded message.
#########################



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