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 <-