> < ^ Date: Fri, 02 Aug 2002 22:43:13 +0300
> < ^ From: Alexander B. Konovalov <alexk@mcs.st-and.ac.uk >
< ^ Subject: Re: Galois group question

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.

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;
```

Sincerely yours,
Alexander Konovalov

Dear GAP Forum,

```Q16 is SmallGroup(16,9)
Q32 is SmallGroup(32,20)
Q24 is SmallGroup(24,6)
```

everything is clear. But now

```QD16 is SmallGroup(16,8)
QD32 is SmallGroup(32,19)
QD24 is ???
```

If someone could please illustrate the answer with an accompanying GAP
command, I'd really appreciate it.

Thanks

Igor

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