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