#### Bruce MacKenzie

##### Member

- Joined
- Sep 3, 2017

- Messages
- 49

I represent a cube state as a permutation of the 48 facelets moved by the cube turns numbered in the order:

UF UR UB UL DF DR DB DL FR FL BR BL UFR URB UBL ULF DRF DFL DLB DBR

The Up facelet of the UP-RIGHT cubie is numbered 1 through to the Right facelet of the Down-Back-Right cubie numbered 48. Numbered in this way the face turns are defined in GAP as:

R := (3,17,11,21)(4,18,12,22)(25,39,46,30)(26,37,47,28)(27,38,48,29);

U := (1,3,5,7)(2,4,6,8)(25,28,31,34)(26,29,32,35)(27,30,33,36);

F := (1,20,9,18)(2,19,10,17)(25,35,40,38)(26,36,41,39)(27,34,42,37);

L := (7,23,15,19)(8,24,16,20)(31,45,40,36)(32,43,41,34)(33,44,42,35);

D := (9,15,13,11)(10,16,14,12)(37,40,43,46)(38,41,44,47)(39,42,45,48);

B := (5,22,13,24)(6,21,14,23)(28,48,43,33)(29,46,44,31)(30,47,45,32);

A cube state defined by a turn sequence may then be entered into GAP as a product of these generators:

1 R U L U' R' U D R D' L' D R' U' D'

g1 := D^-1 * U^-1 * R^-1 * D * L^-1 * D^-1 * R * D * U * R^-1 * U^-1 * L * U * R;

(25,35)(26,36)(27,34)(37,42)(38,40)(39,41)

2 U2 R U2 R' B2 R U2 F2 L' D2 L F2 R' U2

g2 := U^2 * R^-1 * F^2 * L * D^2 * L^-1 * F^2 * U^2 * R * B^2 * R^-1 * U^2 * R * U^2;

(25,30)(26,28)(27,29)(37,47)(38,48)(39,46)

Note that group element multiplication proceeds right to left so the order of the turns is reversed.

In my previous post we were only interested in the position permutation. Numbering the 20 cubies in the order listed above the position permutations of the generating face turns are:

R := (2,9,6,11)(13,17,20,14);

U := (1,2,3,4)(13,14,15,16);

F := (1,10,5,9)(13,16,18,17);

L := (4,12,8,10)(15,19,18,16);

D := (5,8,7,6)(17,18,19,20);

B := (3,11,7,12)(14,20,19,15);