I wish to withdraw, for the time being, my proposed answer to Dan
Hoey's question about how large is the cube group when symmetries
are taken into account. Notwithstanding two rounds of "correction",
I believe my proposal is fundamentally incorrect, and it will take
some time to come up with something better.

I believe that my proposed approximation is incorrect by a factor
of 24. That is, my proposed approximation would be correct for
corners plus edges (without centers), but would need to be
multiplied by 24 in order to be correct for corners plus edges
(with centers). My proposed approximation was
4.3 * 10^19 / (24*24*2). I now believe it should be
4.3 * 10^19 / (24 * 2), with the former figure correct only if
centers are omitted.

Secondly, I believe that my proposed procedure to calculate an
exact value from the known sizes of corner and edge groups is
incorrect. My procedure would be correct if all equivalence classes
had exactly 1152 elements. But they don't. It is not presently
clear to me whether the size of the equivalence classes when
corners and edges are combined can be calculated from the known
sizes of equivalence classes for corners and edges separately,
or whether a computer search will have to be performed for the
case where corners and edges are combined.

I will get back to this in a week or two. In the meantime, my
apologies if I have wasted your time, and I look forward to
any words of wisdom that any of you all might have.

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax
837 Chestnut Ridge Road                     BRYAN@WVNVM
Morgantown, WV 26505                        BRYAN@WVNVM.WVNET.EDU

If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?