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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.
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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).