Date: Wed, 24 Sep 2003 10:49:42 -0400 (EDT)
From: Dan Hoey <Hoey@aic.nrl.navy.mil>
To: John Conway <conway@math.princeton.edu>
Subject: Re: [math-fun] non-square products of squares?
cc: math-fun <math-fun@mailman.xmission.com>

I wrote:
> Up to order 255, the only group that exceeds 1/6 is
> SmallGroup(48,50), or

>     <a,b,c : a^3 = b^2 = c^2 = (a b)^3 = (a c)^3 = (b c)^2
>                  = (a b a^-1 c)^2 = 1> .

> It has a probability of 5/24 that a,b,ab are square,square,nonsquare.

I looked at your classification of groups of order 16 and verified
your translation of the smaller anonymous groups I found.  In hopes of
identifying this group, I found its derived subgroup 2^4 .  Of course
G/G' is C3.  But the direct product C3 x 2^4 is not G.

That seems similar to the case for A4: A4/C3 = 2^2 but 2^2x3 is not
A4.  Somehow I thought that was the way to decompose and reconstitute
groups, but I guess I'm mistaken.

As for identifying the group, I have checked that its order spectrum
[1,15,32,0,...,0] is unique.   Its center is 1.

Dan