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The definition of symmetry proposed by you does not seem adequate.

Plummer's Cross is NOT symmetrical by that definition. Given

Plummer's Cross in any orientation, there are plenty of elements

of the 24-move whole-cube rotation group which will shift it to

something NOT isomorphic through a substitution of colors to the

original. Yet, we know, intuitively, that the CP is "highly

symmetric", and it is a local maximum. It is as though I were

saying that this equivalence of all twists "out" of this position

almost defined symmetry.....