Symmetry and Antisymmetry

You can search for cubes of all types of symmetry/antisymmetry 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 use the symmetry group M of the cube with 48 elements to construct a group with 96 elements using the direct product M x C2, where C2 = {1,a} is the cyclic group of order 2. Now we are able to extend the concept of symmetry to the concept of antisymmetry.

While applying (m ,1) to a cube c just means that we apply the symmetryoperation m to a cube, (m,a) means that we apply m first and then take the inverse of this cube. We call a cube c which is invariant under (m,a) antisymmetric. This is equivalent to the statement that applying m gives the inverse of the cube.
In a mathematically precise sense we define a group action of M x C2 on the set of cubes by
(m,1).y = m y m-1   and   (m,a).y = ( m y m-1)-1 for any cube y. All cubes in the same orbit are related by conjugacy by whole-cube symmetry or/and inversion. In particular all cubes in an orbit share the same optimal maneuver length.

The inverse of a cube has nothing to do with the point reflection at the center of the cube which also is called an inversion. With the inverse of a cube c we mean the inverse regarding the permutation. The inverse of a cube generated for example with the maneuver R U' L2 then is L2 U R'.

While M has 33 different subgroups up to conjugation M x C2 has 131 different subgroups. This can be verified easily with GAP:

M:= Group((1,2,3)(4,6,5),(2,3,5,4),(1,6)(2,5)(3,4));
M1:=DirectProduct(M,Group((1,2)));
Size(ConjugacyClassesSubgroups(M));
33
Size(ConjugacyClassesSubgroups(M1));
131

Each of the 131 subgroups of M x C2 defines an unique type of symmetry/antisymmetry. There are 3 different types of subgroups:

  1. If H is any of the 33 essentially different subgroups of M, (H,1) is a subgroup of M x C2 which is isomorphic to H.
  2. If H is a subgroup of M, (H,1)∪(H,a) also is a subgroup of M x C2. This gives 33 additional cases.
  3. If H1 and H2 are subgroups of M with H1⊂ H2 and | H2 | = 2 | H1 |, then (H1,1) ∪( H2\ H1,a) is a subgroup of M x C2. H2\ H1 denotes the difference set of H2 and H1.We have 65 different types of this most interesting case.

Symmetries/Antisymmetries of type 3

In the animated gifs below, the blinking elements are those of H2\ H1. Applying these elements of M give the inverse cube, while applying the not blinking elements of H1 leave the cube unchanged.

Type H
H1
H2
Subgroup size |H|
Size of conjugacy class
Number of cubes
mod M x C2with exactly this symmetry
Shortest generator for exactly this antisymmetry
3.1
O
Oh
48
1
0
---
3.2
Td
Oh
48
1
0
---
3.3
Th
Oh
48
1
4
B2 L R2 B2 F2 D2 U2 R' F2 D U2 B2 F2 L2 R2 U' (16f*)
3.4
T
O
24
1
6
B F L' R' B F D' U' L R D' U' (12f*)
3.5
T
Td
24
1
6
B F L R B' F' D' U' L R D U (12f*)
3.6
C3v
D3d
12
4
8
D U R2 B D2 U2 L' D2 U2 R B' R2 D' U' (14f*)
3.7
D3
D3d
12
4
88
D B D U2 B2 F2 L2 R2 U' F U (11f*)
3.8
S6
D3d
12
4
1338
B' D' U L' R B' F U (8f*)
3.9
T
Th
24
1
0
---
3.10
C3
C3v
6
4
43740
D' B2 D L2 D2 B2 U B2 U' (9f*)
3.11
D4
D4h
16
3
80
U D (2f*)
3.12
D2d (face)
D4h
16
3
80
U' L R B2 F2 L' R' U' (8f*)
3.13
C4v
D4h
16
3
176
F2 R2 D2 F2 L R U2 F2 L' R' (10f*)
3.14
C4h
D4h
16
3
448
U D' (2f*)
3.15
D2h (edge)
D4h
16
3
0
---
3.16
D2d (edge)
D4h
16
3
496
B F L2 R2 B' F' U2 (7f*)
3.17
D2h(face)
D4h
16
3
60
B2 L R2 B2 F2 D2 U2 R' F2 D' B2 F2 L2 R2 U' (15f*)
3.18
C3
D3
6
4
8780
B2 D' U' R2 B2 U2 F2 D U' (9f*)
3.19
D2 (edge)
D4
8
3
64
L R B' F' U2 B' F' U2 L' R' (10f*)
3.20
C4
D4
8
3
2080
U B2 F2 L2 R2 D' (6f*)
3.21
D2 (face)
D4
8
3
490
B' F' L R (4f*)
3.22
C2v (a1)
D2d (face)
8
3
32
L2 B2 U2 B2 R2 U2 L2 D' U R2 (10f*)
3.23
D2 (face)
D2d (face)
8
3
490
B F L R (4f*)
3.24
S4
D2d (face)
8
3
6144
D' B2 F2 U2 B2 F2 U' (7f*)
3.25
C2v (a1)
C4v
8
3
32
U B2 F2 D2 L2 R2 U' (7f*)
3.26
C4
C4v
8
3
4192
U (1f*)
3.27
C2v (a2)
C4v
8
3
1136
L2 R2 D2 B2 F2 (5f*)
3.28
C4
C4h
8
3
448
B F U B F L R D L R (10f*)
3.29
S4
C4h
8
3
1568
B' F' D' B' F' L R D L R (10f*)
3.30
C2h (a)
C4h
8
3
0
---
3.31
C2v (a1)
D2h (edge)
8
3
4544
B2 F2 U R2 B2 F2 R2 U' L2 R2 (10f*)
3.32
C2v (b)
D2h (edge)
8
6
12800
D B2 D' U' B2 D2 U' (7f*)
3.33
C2h (b)
D2h (edge)
8
6
12800
D' R2 D' U' R2 U' (6f*)
3.34
D2 (edge)
D2h (edge)
8
3
6688
D' F2 D2 U2 F2 U' (6f*)
3.35
C2h (a)
D2h (edge)
8
3
10176
R2 D2 U2 R2 D U' (6f*)
3.36
D2 (edge)
D2d (edge)
8
3
144
L2 F' L2 D2 U2 R2 B' R2 D U (10f*)
3.37
S4
D2d (edge)
8
3
6224
U L R U2 L R U (7f*)
3.38
C2v (a2)
D2d (edge)
8
3
192
D' L2 B2 U2 B2 R2 U2 L2 D' U R2 U' (12f*)
3.39
C3
S6
6
4
364
U' F2 R2 D U2 F2 D2 L2 U' B2 R2 U' (12f*)
3.40
D2(face)
D2h(face)
8
1
2288
D B2 D2 U2 F2 U (6f*)
3.41
C2h (a)
D2h(face)
8
3
14144
D' B2 D2 U2 F2 U (6f*)
3.42
C2v (a2)
D2h(face)
8
3
29232
U2 L2 B2 F2 R2 (5f*)
3.43
Cs (b)
C2v (a1)
4
6
481856
B' L B L' R F' R' F (8f*)
3.44
C2 (a)
C2v (a1)
4
3
1,776,960
B F U L R (5f*)
3.45
Cs (b)
C2v (b)
4
6
481856
R D' R' D B R' (6f*)
3.46
C2 (b)
C2v (b)
4
6
2,024,512
U R2 F2 D U' R2 U (7f*)
3.47
Cs (a)
C2v (b)
4
6
3,539,648
R2 B2 (2f*)
3.48
Cs (b)
C2h (b)
4
6
481848
U' R2 D B2 R2 B2 D B2 U' (9f*)?
3.49
C2 (b)
C2h (b)
4
6
2,024,424
D' R2 D2 B2 D2 R2 U' (7f*)
3.50
Ci
C2h (b)
4
6
3,695,238
L' R B' F (4f*)
3.51
C2 (b)
D2 (edge)
4
6
592064
U R2 U2 F2 D2 R2 U' (7f*)
3.52
C2 (a)
D2 (edge)
4
3
514384
L R D' U B' F' (6f*)
3.53
C2 (a)
C4
4
3
10832
B' F' U2 L R (5f*)
3.54
C2 (a)
D2(face)
4
3
512816
D L2 R2 U' (4f*)
3.55
C2 (a)
S4
4
3
3264
U B' F' L' R' D (6f*)
3.56
C2 (a)
C2h (a)
4
3
1,765,392
D L R U (4f*)
3.57
Cs (a)
C2h (a)
4
3
1,731,088
L2 B2 L2 D2 U2 (5f*)
3.58
Ci
C2h (a)
4
3
1,863,344
D2 L2 B2 L2 U2 (5f*)
3.59
C2 (a)
C2v (a2)
4
3
1,766,720
D L R D (4f*)
3.60
Cs (a)
C2v (a2)
4
6
3,474,976
F2 U2 B2 (3f*)
3.61
C1
Cs (b)
2
6
108,272,809,188
R B (2f*)
3.62
C1
C2 (b)
2
6
21,419,485,172
R B' (2f*)
3.63
C1
C2 (a)
2
3
10,677,084,112
F U2 B' (3f*)
3.64
C1
Cs (a)
2
3
54,180,798,352
U R D (3f*)
3.65
C1
Ci
2
1
18,059,430,572
R U R L D L (6f*)

Symmetries/Antisymmetries of type 2

Cubes having this symmetry/antisymmetry are those of antisymmetry type 1 with the additional requirement that they are selfinverse.

 
Type H
Schoenflies-Symbol
Subgroup size |H|
Size of conjugacy class
Number of cubes
mod M x C2with exactly this symmetry
Shortest generator for exactly this antisymmetry
2.1
Oh
96
1
4
Solved Cube (0f*)
2.2
O
48
1
0
---
2.3
Td
48
1
0
---
2.4
Th
48
1
2
U2 L2 F2 D2 U2 F2 R2 U2 (8f*)
2.5
T
24
1
0
---
2.6
D3d
24
4
12
U L D U L' D' U' R B2 U2 B2 L' R' U' (14f*)
2.7
C3v
12
4
8
U L' R' B2 U' R2 B L2 D' F2 L' R' U' (13f*)
2.8
D3
12
4
24
D' F' U B F' R' B L R' D' L U (12f*)
2.9
S6
12
4
12
U2 R2 U B2 F' D' U' F' D U B2 F' U' R2 U2 (15f*)
2.10
C3
6
4
164
B U' L F U L' B' U2 F R' F2 U2 (12f*)
2.11
D4h
32
3
124
D2 U2 (2f*)
2.12
D4
16
3
80
U B2 F2 L2 R2 U' (6f*)
2.13
C4v
16
3
176
U2 (1f*)
2.14
C4h
16
3
64
B2 D U' R2 F U2 F R' D2 L' B2 D2 B' L' F R' (16f*)
2.15
C4
8
3
448
B2 F2 U L2 R2 D2 B2 F2 U' L2 R2 (11f*)
2.16
S4
8
3
368
B2 R' D2 B2 F2 U2 L' F2 U2 (9f*)
2.17
D2d (edge)
16
3
336
U' L2 R2 U2 L2 R2 U' (7f*)
2.18
D2d (face)
16
3
80
U L2 B' L2 D2 U2 R2 F' R2 U' (10f*)
2.19
D2h (edge)
16
3
960
U' B2 F2 D2 L2 R2 U' (7f*)
2.20
D2h(face)
16
1
1302
B2 D2 U2 F2 (4f*)
2.21
D2 (edge)
8
3
3312
U R2 D2 U2 R2 U' (6f*)
2.22
D2 (face)
8
1
880
B L' R B2 F2 L R' F' (8f*)
2.23
C2v (a1)
8
3
4544
L2 B' L2 D2 U2 R2 F' R2 U2 (9f*)
2.24
C2v (a2)
8
3
29312
B2 D2 U2 F2 U2 (5f*)
2.25
C2v (b)
8
6
12800
D' L2 D U L2 U' (6f*)
2.26
C2h (a)
8
3
13856
U' B F' L2 R2 B F' U (8f*)
2.27
C2h (b)
8
6
12788
D U2 R2 D' U' R2 U' (7f*)
2.28
C2 (a)
4
3
512640
D L2 R2 D' (4f)
2.29
C2 (b)
4
6
592040
U R2 U2 F2 U2 R2 U' (7f*)
2.30
Cs (a)
4
3
1,731,088
R2 F2 R2 (3f)
2.31
Cs (b)
4
6
481848
B2 D' F2 L2 B2 U' R2 (7f*)
2.32
Ci
4
1
616848
R2 U2 F2 U2 R2 (5f)
2.33
C1
2
1
3,558,670,020
R U2 R' (3f)

Symmetries/Antisymmetries of type 1

These symmetries/antisymmetries directly correspond to the 33 symmetry types of the cube given here. So it might be better to call cubes which are invariant under one of these subgroups of M x C2 symmetric instead of antisymmetric. But be aware that there is a subtle difference between a cube having the symmetries of a subgroup H of M and a cube having the symmetries of a subgroup (H,1) of M x C2. For example entry 1.5 tells us that there are no cubes with symmetry (T,1). But there are 12 cubes which exactly have the symmetry of the subgroup T of M if we ignore antisymmetries.

 
Type H
Schoenflies-Symbol
Subgroup size |H|
Size of conjugacy class
Number of cubes
mod M x C2with exactly this symmetry
Shortest generator for exactly this antisymmetry
1.1
Oh
48
1
0
---
1.2
O
24
1
0
---
1.3
Td
24
1
0
---
1.4
Th
24
1
2
R2 D2 R B2 D U F2 R U2 R2 D' F2 L' R' F2 U' (16f*)
1.5
T
12
1
0
---
1.6
D3d
12
4
0
---
1.7
C3v
6
4
0
---
1.8
D3
6
4
48
R F R' U R D B D U2 B2 F2 L2 R U' (14f*)
1.9
S6
6
4
1260
D' U' B2 L2 D' U R2 F2 U2 (9f*)
1.10
C3
3
4
444834
D' L B2 L' B' U B2 U' B D (10f*)
1.11
D4h
16
3
0
---
1.12
D4
8
3
16
D U B2 F2 L2 R2 (6f*)
1.13
C4v
8
3
48
U2 B2 F2 L2 R2 (5f*)
1.14
C4h
8
3
96
R2 B R2 D' B2 F2 L2 R2 U' R2 B' R2 (12f*)
1.15
C4
4
3
14496
U F2 B2 R2 L2 (5f*)
1.16
S4
4
3
47536
U' F' B' U2 R2 L2 F' B' U (9f*)
1.17
D2d (edge)
8
3
320
F2 L2 R2 U2 L2 R2 U2 F2 U2 (9f*)
1.18
D2d (face)
8
3
16
D' U' R2 B2 F2 U2 B2 F2 U2 R2 (10f*)
1.19
D2h (edge)
8
3
0
---
1.20
D2h(face)
8
1
310
B' L2 R2 D2 U2 F' U2 L' B2 F2 D2 U2 R' U2 (14f*)
1.21
D2 (edge)
4
3
6512
U' R' L' F' B' R L U (8f*)
1.22
D2 (face)
4
1
9604
R L U2 D2 (4f*)
1.23
C2v (a1)
4
3
3200
R2 L2 U' F2 B2 D' F2 R2 L2 F2 (10f*)
1.24
C2v (a2)
4
3
115504
U2 L2 R2 (3f*)
1.25
C2v (b)
4
6
11264
B' U' F2 D L2 D R2 U' B2 F R2 L2 (12f*)
1.26
C2h (a)
4
3
52688
R2 L2 U D' (4f*)
1.27
C2h (b)
4
6
11264
B2 R2 U' F2 L2 B' D' B R2 F' U F (12f*)
1.28
C2 (a)
2
3
951,909,808
U R L (3f*)
1.29
C2 (b)
2
6
315,851,984
U L2 D U L2 U' (6f*)
1.30
Cs (a)
2
3
1,141,184,640
U D' R2 (3f*)
1.31
Cs (b)
2
6
52,088,192
D R2 D U R2 U' (6f*)
1.32
Ci
2
1
952,378,138
R L U2 F2 U2 R2 (6f*)
1.33
C1
1
1
450,541,590,977,171,858
U R2 (2f*)

How to count the number of "essentially" different cubes up to symmetry and inversion

In 2005 Mike Godfrey and I obtained this number using the Lemma of Burnside. A direct analysis of the 131 different subgroup types not only verifies the number 450,541,810,590,509,978 but furthermore gives the number of essentially different cubes having a specific symmetry type H.

For any cube y, the stabilizer subgroup Stab(y) is the subgroup H of M x C2 that fixes y and hence defines the symmetry/antisymmetry of y.

The orbit Orb(y) for any cube y is the set of all cubes we get by applying the group action of all 96 elements of M x C2 to y. All cubes in Orb(y) are equivalent, in particular they have the same maneuver length.

Let S(H) be the union of all cubes which have the stabilizer subgroup H or a stabilizer subgroup conjugate to H. Then the orbit Orb(y) of any element of S(H) is a subset of S(H). Each Orbit has the same number of elements, namely |M x C2| / |H| = 96/|H|.

If we divide |S(H)| by this number we get the number O(H) of different orbits of S(H). We can interpret O(H) as the number of essentially different cubes which have the symmetry/antisymmetry of type H.

O(H) = |S(H)| |H| / 96

Though it seems difficult to classify all 43,252,003,274,489,856,000 different cubes regarding their stabilizer subgroup on first sight, separating corners and edges and the fact that most of the cubes are of type 1.33 make it possible to do this classification within a couple of hours of CPU-time. O(H) is given in the column "Number of cubes mod M x C2 with exactly this symmetry".

If we add the counts O(H) for all 131 cases we exactly get 450,541,810,590,509,978 in accordance with the result of 2005.