The Group D₂ (edge)

Silviu Radu did most of the analysis of the 98304 different cubes of this subgroup. Here the the table with the distances, the average distance is 16.93:

Distance	Number	Distance	Number
0f	1	11f	1030
1f	0	12f	1844
2f	3	13f	2134
3f	0	14f	4604
4f	1	15f	5959
5f	0	16f	11355
6f	28	17f	22676
7f	44	18f	36221
8f	128	19f	11617
9f	71	20f	94
10f	494

Up to M-symmetry there are 23232 cubes which exactly have this symmetry. All cubes can be solved within 20 moves and there are 12 cubes up to M-symmetry which need 20 moves. All except 2 of these cubes are antisymmetric (that is the inverse is M-symmetric to the cube itself).
Some nice examples are listed below.
The next table gives the number of cubes up to M-symmetry which exactly have D₂ (edge)-symmetry, the average maneuver length is 16.96 now:

Distance	Number	Distance	Number
0f	0	11f	230
1f	0	12f	403
2f	0	13f	461
3f	0	14f	1064
4f	0	15f	1396
5f	0	16f	2667
6f	4	17f	5397
7f	6	18f	8736
8f	22	19f	2713
9f	10	20f	12
10f	111

If you are interested in a list of all optimal maneuvers for this subgroup, they are included in the file D2e.zip. The 11 20f*-maneuvers up to M-symmetry and M-antisymmetry are also included in the file 20moves.zip.

Name	shortest maneuver with exactly this symmetry
Generator	U F2 U2 D2 F2 D (6f*)

Name
Generator	U' R2 D2 U2 R2 D2 U' (7f*)

Name
Generator	R2 U' B2 R' B U' B L2 U L2 B U B' R' (14f*)

Name
Generator	L R' F2 U2 B' L2 R2 F D2 F2 L R' (12f*)

Name
Generator	L B2 D2 L F' D2 U2 B R' D2 F2 R' (12f*)

Name
Generator	U' R' F2 R2 F2 D U F2 R2 F2 R2 D' U' R' U (15f*)

Name
Generator	R' U' B' F2 D' L B' L2 F' L D' B2 F' U' R' U2 (16f*)

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The Group D2 (edge)

The Group D₂ (edge)