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mark writes

[ ... ]

The following command shows that pons is at the centre of slice group:

Size (Centralizer (slice, pons)) = 768

this is not hard to see. the center of the slice group has order 32.

if we hold the corners fixed, then these central elements are exactly

those for which the six face-centers are correct.

[ ... ]

Of course, now that I have answered my old questions, I must

formulate new ones....A) What is the next most commutative element (the pancentre?)

after the 12-flip?

B) What is the least commutative element (the anticentre?) of

the cube group?

i'm sure that GAP can do these. you're interested in knowing about

the orders of centralizers of various elements. for an element

g in a group G , we have

|G| / |Z(g)| = number of conjugates of g .

this is because the cosets G / Z(g) are in one-to-one correspondence

with the conjugates of g. of course, this doesn't help much unless

we know about conjugacy classes in G.

in the case of the cube group, however, conjugacy classes are easy to

understand. they are (almost) completely described by cycle structure.

(some cycle structures have two conjugacy classes.) there are many

different possible cycle structures, but for each it should be easy to

count the number of elements with that cycle structure (and also to

tell whether they comprise a single conjugacy class or split into two).

mike