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Okay, I understand the GAP conventions better now.

If we adhere to the following model for the cube:

+--------------+ | 1 2 3 | | 4 top 5 | | 6 7 8 | +--------------+--------------+--------------+--------------+ | 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 | | 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 | | 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 | +--------------+--------------+--------------+--------------+ | 41 42 43 | | 44 bottom 45 | | 46 47 48 | +--------------+

Then the Pons Asinorum would be: pons := ( 2,42)( 4,45)( 5,44)( 7,47)(10,31)(12,28)(13,29)(15,26) (18,39)(20,36)(21,37)(23,34);; And the slice group would be: slice := Group( ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19) (41,46,48,43)(42,44,47,45)(14,38,30,22)(15,39,31,23)(16,40,32,24), ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35) (25,30,32,27)(26,28,31,29)( 3,19,43,38)( 5,21,45,36)( 8,24,48,33), (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11) (33,38,40,35)(34,36,39,37)( 3,32,46, 9)( 2,29,47,12)( 1,27,48,14) );; And the anti-slice group would be: antisl := Group( ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19) (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40), ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35) (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24), (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11) (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27) );; Size (antisl) = 6,144 Size (slice) = 768

These numbers concur with Mr. Singmaster's earlier "Notes".

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

Size (Centralizer (slice, pons)) = 768

Once again, I will refer to Martin's earlier statement about

centralizers:

That is, of the total 980995276800 elements in GE only 980995276800/332640 = 2949120 elements centralize P. And I used the definition of P from your e-mail of 1995/01/03, i.e., P = (F2 B2) (U2 D2) (L2 R2) = (F2 B2) (L2 R2) (U2 D2) = ... (one gets the same element independent of the order of the three pairs).

So now that I have the groups and pons element correct:

Size (Centralizer (edge, pons)) = 2,949,120

I wrote some statements before....

Only 2,949,120 elements of GE centralize P,

also only...

2,949,120 elements of G centralize P

I am only partly correct as....

Size (Centralizer (cube, pons)) = 130,026,464,870,400

As Martin said before:

> Only one out of 332640 elements of GE (and of G) centralizes P.

Size (cube) / 332640 = 130,026,464,870,400 or 130 trillion and change.

...the full cube group has many more elements which commute with

pons than the mere edge group!

GAP is a very function-laden beastie:

Size (Intersection (antisl, slice)) = 8

This function gives the number of elements included in both the

anti-slice and slice groups.

Naturally there is a corresponding Union function.

Since I have studied the squares group and the <U, R> group, the

number of elements in the intersection of the two are of

particular interest:

Size (Intersection (ur, sq)) = 72 And now we have a new way to check an old result :-) Order (cube, uturn * rturn) = 105

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?

-> Mark <-