The Group Th

Up to M-symmetry there are 10 cubes, which exactly have the symmetries of this subgroup. 4 of these cubes need 20 moves to be generated.

The next table gives the number of cubes up to M-symmetry which exactly have Th-symmetry, the average maneuver length is 16.9:

Distance
Number
Distance
Number
8f
1
17f
1
16f
4
20f
4

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file Th.zip.The three 20f*-maneuvers up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.
Name
 6 H's, shortest maneuver with exactly this symmetry
Generator
 U2 L2 F2 D2 U2 F2 R2 U2 (8f*)
Name
 Supertwist
Generator
 B2 L R2 B2 F2 D2 U2 R' F2 D U2 B2 F2 L2 R2 U' (16f*)
Name
 Supertwist composed with 6 H's
Generator
 R2 D2 R B2 D U F2 R U2 R2 D' F2 L' R' F2 U' (16f*)
Name
 Supertwist composed with Pons Asinorum
Generator
 B2 U' R2 B F R2 D F2 D2 B R2 D U R2 F' U2 (16f*)
Name
 Supertwist composed with 6 H's and Pons Asinorum
Generator
 L F2 D U F2 R U2 L2 U' F2 L' R' B2 U' R2 U2 (16f*)
Name
 6 H's composed with Superflip
Generator
 U L' B' L' R U F L' R U' R' D' U B' F2 R U' (17f*)
Name
 Superfliptwist
Generator
 B U' R D2 B' D2 B' R B' D2 F U R F L B2 D' U' R' U' (20f*)
Name
 Superfliptwist composed with 6 H's and Pons Asinorum
Generator
 U F2 R2 F2 D U2 B U' L' R F' U' F2 L' D2 B' R2 D F2 R' (20f*)
Name
 Superfliptwist composed with 6 H's
Generator
 B2 R' B2 F U L2 F2 L F' D B F2 L2 F2 R U L2 B' R' U' (20f*)
Name
 Superfliptwist composed with Pons Asinorum
Generator
 B' F U L U2 B L2 B2 R2 D' L' R D L2 F2 R2 F' U2 R' U' (20f*)

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