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