Date: Sun, 24 Sep 95 22:56:00 -0400
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Least Commutative Element

# The 2nd most commutative element of the cube group
# It is the position of 7 clockwise and 1 counter-clockwise twists
# This commutes with 1 out of every 8 elements of the cube

```commuter := ( 1,35, 9)( 3,27,33)( 6,11,17)( 8,19,25)(24,43,30)(32,48,38)
(14,40,46)(16,22,41);;

# The least commutative element of the cube group ( I think! )
# This commutes with 1 out of every 450,541,700,775,936,000
#  or approximately 1 out of every 4.5 * 10^17 patterns

least := ( 1, 6,32,19, 3,41,24,46)( 2, 4,13,42,29, 7,31,15,39,37,26,21)
( 5,28,34,10,20,23,36,18,45,44,47,12)( 8,33,16,30,40, 9,17,38)
(11,48,25,27,22,43,14,35);;
```

after thinking about it, i realized that

```corners:  (8)   edges:  (12)
```

commutes with even fewer elements. again, elements with
this cycle structure split into two conjugacy classes.

mike

With GAP we must deal with permutations of cube facelets, and that is
why the permutation 'least' has 3 sets of 8 numbers and 2 sets of
12 numbers. Moreover, as I'm sure Mike will appreciate, the least
commutative element I've found is has a 8-cycle of corners and a
12-cycle of edges.

Size (Centralizer (cube, least));
96

```-> Mark <-
```