Jerry Bryan writes:

I don't yet understand why Mike's position is a local maximum in the
full cube group. But assuming it is, it is not only the shortest
local maximum, it is the first local maximum which is not
Q-transitive (i.e, we have |{m'Xm}|=24, hence we have |Symm(X)|=2,
and the size of the symmetry groups for Q-transitive positions
must be divisible by 12.).

No, the 4-spot pattern is also a local maximum at 12 qtw, although its
symmetry group is of order 16. Jim Saxe and I reported this on 22
March 1981, in "No short relations and a new local maximum".

Dan
Hoey@AIC.NRL.Navy.Mil