Optimal Sequences for < U, R > group elements (positions)
---------------------------------------------------------
Edge 3-cycle
UR1 = U3 R1 U2 (R1 U1)^2 R2 U3 R3 U3 R2 U1 (16 q, 13 h)
UR1a= F1 U2 (F1 U1)^2 F2 U3 F3 U3 F2 (14 q, 11 h)
UR1b= F3 U1 F3 (U3 F3)^2 U1 F1 U1 F2 (12 q, 11 h)
Double adjacent edge swap
UR2 = U2 R3 U3 R2 U1 R1 U1 R3 U3 R1 U1 (R1 U3)^2 R3 (18 q, 16 h)
Diagonal Corner twist
UR3 = U1 R1 U3 R1 U3 R2 U1 R1 U1 R3 (U3 R1)^2 U2 R3 U3 R3 (20 q, 18 h)
Double opposite edge swap, also in sq group 24 q, 12 h
UR4 = R2 U2 R3 (U2 R2)^2 U2 R3 U2 R2 (20 q, 11 h)
Edge 7-cycle, equivalent to (U1 R1)^15
UR5 = U3 R1 U3 R3 U3 R1 U2 R3 U1 R3 U2 R1 U3 R3 (U3 R1)^2 (20 q, 18 h)
Corner Tri-Twist
UR6 = (U3 R3)^2 U1 R1 U3 R3 U3 R2 U1 R2 U3 R3 U3 R1 U1 R3 (20 q, 18 h)
Corner Quad-Twist, Flat style
UR7 = R1 U3 (R1 U1)^2 (R3 U3)^2 R2 U3 R1 U1 R3 U3 R1 U3 R3 (20 q, 19 h)
Corner Quad-Twist, Arms & Legs style
UR8 = R1 U1 R3 U1 R3 U3 R1 U1 R1 (U3 R3)^2 (U1 R1)^2 U3 R3 U3 (20 q, 20 h)
ML Max Position
UR9 = (U2 R2)^2 U2 R3 U1 R2 (U3 R2)^2 U1 R1 (22 q, 14 h)
Same position found by hand: (a non-optimal 24 q, 15 h)
(U2 R2)^3 U1 R1 (U2 R3)^2 U2 R1 U1
4 Opp Corner Swap, also in sq group at 26 q, 13 h
UR10 = U3 R3 (U1 R1)^2 U2 R3 U1 R1 (U2 R2)^2 U1 R3 U1 (22 q, 17 h)
6 Twist, Equivalent to (U1 R1)^35= (R1 U1)^35 & Shift Invariant (22 q, 20 h)
UR11 = (U2 R1 U1 R1 (U1 R3)^3 )^2
6 Twist + 4 Opp Corner Swap or UR10 + UR11 (22 q, 19 h)
UR12 = U1 R3 U3 R3 U1 R1 U1 R3 U3 R1 U2 (R2 U1 R3 U3 R2 U3 R1 U1)
First Antipodal Process (25 q, 20 h)
UR13 = U2 R1 (U3 R2)^2 U3 R1 U2 R3 U1 R3 U3 R1 U1 R2 (U1 R3)^2
First Q+H Antipodal Process (24 q, 20 h)
UR14 = (U3 R3)^4 U1 R2 U1 (R3 U2)^2 R3 U1 R2 U1 R3
Other Subgroups within reach
----------------------------
11. |__| = 2^12 3^4 5^2 7 = 58060800
12. || = 2^12 3^4 5^2 7 = 58060800
17. |____| = 2^8 3^5 5^2 7 = 10886400
21. |____| = 2^13 3^4 5^2 7 = 116121600
22. |____| = 2^15 3^4 5^2 7^2 = 3251404800
__