I said:
> Still trying to find a pattern which will
> result in 4 distinct ways, but I am not optimistic.

Jerry adds:
> As one more followup, for each symmetry group order in the above list,
> there exists at least one cube.
> That is, 96 of the 98 subgroups are symmetry groups for at
> least one cube. The two "missing" subgroups -- A and C -- are of
> order 24. But there is a third subgroup -- H -- of order 24
> (H is the set of 12 even rotations and 12 odd reflections), and there
> are cube positions whose symmetry subgroup is H. Hence, there are
> cube positions for every symmetry subgroup order.

Well, I figure Jerry is correct and so I kept looking for the magic
pattern which transforms 4 ways...

Number of
different      Pattern
patterns        -------
---------

...
 4              6 flip (UF, UR, FR, DB, DL, BL)
...

So there are 4 types of this 6 flip.

Jerry has said before:
> I believe that Dan and I have solved (sort of independently, and sort
> of working together) the problem you pose (and I give Dan the bulk
> of the credit). That is, how many cubs are there in each symmetry
> group and each symmetry class?

That sounds harder. Looks like I am specifying only the index of
the symmetry subgroup... perhaps it makes sense to find out
exactly which subgroup of M is the symmetry group of my positions.

It all sounds vaguely familar.... but it will try again tomorrow.

-> Mark <-