Symmetric Patterns in Detail

You can search for cubes of all symmetry types with the Symmetry Editor module of Cube Explorer.

Look here first for the mathematical background of symmetric patterns and an explanation of the pictograms.

An external page with good information about the schoenflies symbols can be found here.

We know God's algorithm for all the 164,604,041,664 symmetric cubes which exist. The following table gives the distribution:

Distance
Number
Distance
Number
0f
1
11f
9,732,164
1f
18
12f
35,024,904
2f
51
13f
122,054,340
3f
312
14f
436,197,214
4f
1,335
15f
1,763,452,505
5f
4,380
16f
8,035,307,127
6f
17,782
17f
37,542,012,922
7f
70,188
18f
95,387,902,305
8f
229,336
19f
21,267,102,443
9f
851,139
20f
1,091,994
10f
2,989,204
21f
0

Reducing the 1,091,994 symmetric cubes with 20 moves by symmetry and antisymmetry we find exactly 32,625 essentially different symmetric cubes which need 20 moves to be solved. They are included in the file 20moves.zip.

The details for the different symmetry types can be found below.

 

Type
Schoenflies-Symbol
Number of Symmetries
Number of cubes having at least this symmetry
Shortest generator for exactly this symmetry
More Information
Oh
48
4
do nothing
O
24
4
---
Td
24
4
---
Th
24
24
U2 L2 F2 D2 U2 F2 R2 U2
T
12
72
B F L R B' F' D' U' L R D U
D3d
12
16
U L D U L' D' U' R B2 U2 B2 L' R' U'
C3v
6
48
U L' R' B2 U' R2 B L2 D' F2 L' R' U'
D3
6
432
D B D U2 B2 F2 L2 R2 U' F U
S6
6
7776
B' D' U L' R B' F U
C3
3
3,779,136
L' R U2 R2 D2 F2 L R D2
D4h
16
128
U2 D2
D4
8
512
U D
C4v
8
1024
D2
C4h
8
1536
U D'
C4
4
147456

U

S4
4
442368
U R2 L2 U2 R2 L2 D
D2d (edge)
8
3072
U F2 B2 D2 F2 B2 U
D2d (face)
8
512
U R L F2 B2 R' L' U
D2h (edge)
8
2048
U R2 L2 D2 F2 B2 U
D2h(face)
8
12288
B2 D2 U2 F2
D2 (edge)
4
98304
U F2 U2 D2 F2 D
D2 (face)
4
294912
R2 L2 F B
C2v (a1)
4
65536
U R2 L2 U2 F2 B2 U'
C2v (a2)
4
1,179,648
R2 L2 U2
C2v (b)
4
98304
B2 R2 B2 R2 B2 R2
C2h (a)
4
589824
U' D F2 B2
C2h (b)
4
98304
U R2 U D R2 D
C2 (a)
2
15,288,238,080
L R U2
C2 (b)
2
2,548,039,680
U R2 D' U' R2 U'
Cs (a)
2
18,345,885,696
F2 R2
Cs (b)
2
424,673,280
U B2 U D B2 D'
Ci
2
45,864,714,240
U D' R L'
C1
1
43,252,003,274,489,856,000
U R