I wrote in my e-mail message of 1994/12/07
But C is not the largest such group. The largest such group is M, i.e.,
the full group of symmetries of the entire cube. This is the reason why
I prefer to view G as a subgroup of MG, which is the semidirekt product
of M and G, even though I realize that MG is not physically realizable.
Jerry Bryan answered in his e-mail message of 1994/12/07
But can't you speak of conjugates such as m'gm without regard to G
being a subgroup of MG? I agree that MG seems like a very useful group,
and it is a very nice model of what is going on. But doesn't g in G
imply m'gm in G whether I ever heard of MG or not?
Yes I certainly could. I think it is only a matter of taste.
You seem to favor the physical model. There the reflection has no
real realization, and it makes sense to distinguish between the
rotations and the reflection.
I look at the problem more from the computational aspect. I view
the whole thing as a permutation group, and then there is no real
reason to distinguish between the rotations and the reflection
(both being ordinary permutations on 54 points).
And when working with those groups in GAP, it is certainly a lot more
convenient to work in MG and treat all of M uniformly, then to work in
CG and to handle the reflection specially.
Have a nice day.
-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany