Rubik's Cube symmetries and pretty patterns

25. Symmetry r4, <(1234)>, 4.

Hold the cube so that the r4 symmetry is centred around the U/D faces. Note that this group of patterns includes those of sections 8, 13 and 14.

The edges and corners all fall into orbits of 4 pieces; the U edges, the U corners, the D edges, the D corners, and the middle layer edges. Any of these orbits can move around in a 4-cycle, as long as this odd permutation is compensated for by a another 4-cycle.

4T pattern of order 4; 4-cycle of corners, 4-cycle of edges (URF,UFL,ULB,UBR)(RF,FL,LB,BR)
R' B' Rs F' R' U R F Rs' B R (13f*,13q*)
4K pattern of order 4; two 4-cycles of edges type 3 [p114] (UF,UL,UB,UR)(RF,FL,LB,BR)
Fs L F2 L2 Us B U2 Rs B2 D L2 U' (15f*)
D B L U F' D' B' F L B L' R D' U R' F' R' U' (18q*)

By using these in a different orientation, the D edges/corners can be cycled around as well. The middle layer can be cycled twice on its own by 5i.

The U/D edge orbits can be swapped by 5f, and the corner orbits by 5b. The U edges can be flipped by 14c (ditto the D edges), the middle edges by 5d. The corners orbits cannot be twisted separately, but only together by 13b.

These patterns form a group of order 147,456 (= 2¹⁴3²).

There are two types of patterns. Here are the rest of order 4:

One layer turn (25a+25b+5i)
U (1f*,1q*)
4Y of order 4 [p164] (25a+25b'+5i)
Fs2 Rs2 D' Fs2 Rs2 (9f*,17q*)
(U Rs2)⁶ U
4D of order 4 (25a+25b+25b'+5i)
U L2 D2 Fs' R' F2 Us L2 F' D2 B2 Rs' (15f*)
B L U F' D' B' F L B L' R D' U R' F' R' U' (17q*)
4U of order 4 [p112] (25a+25a'+25b+5i)
F R' B' D2 L' Us B D2 R F L' (12f*,14q*)
2X + 4T pattern of order 4 (25a+5f)
D2 Rs2 B2 D' L2 Fs U2 L' Us' F2 R2 F' Rs' (17f*)
2X + 4K pattern of order 4 (25b+5f)
F2 Rs' D2 F' R2 Us' F2 Us2 L D2 Fs' R2 (16f*)
2X + One layer turn (25a+25b+5i+5f)
U2 Rs2 Us Rs2 D' (8f*,13q*)
2X + 4Y of order 4 [p169] (25a+25b'+5i+5f)
(Fm2 D Rm2)² D
U Rs2 Us' Rs2 U2 (8f*,13q*)
2X + 4D of order 4 (25a+25b+25b'+5i+5f)
U' R2 D F2 L2 D Fs L Us' F2 R2 F U2 Ra' (17f*)
2X + 4U of order 4 (25a+25a'+25b+5i+5f)
R2 D' B2 L2 D' Fs L B2 L2 Us' F U2 Ra (16f*)

If we swap the corner orbits, then we will get 3 colours on the sides unless we use sequence 8b or 13h. Note that 8b = 25a^-1+5b+25a. Now the faces have only opposing colours on them, so we should not apply a 4-cycle unless we do it twice. To break the extra r2 symmetry of 8b, we can assume that we use 25b but not 25b', so this leaves only the following patterns of order 2:

(8b+25b²)
Rs2 F2 L2 Ua B2 R2 Ua' F2 R2 (12f*)
(R2 F2 U2)² F2 R2 D2 (R2 B2)² D2
(8b+25b²+5i)
F2 R2 Ua' B2 L2 Fs2 Ua F2 R2 (12f*)
(F2 R2 U2)² R2 F2 D2 (F2 R2)² D2
(13h+25b²)
F2 U2 F2 Ra' (F2 U2)² F2 U2 Ra (11f*)
(13h+25b²+5i)
B2 U2 B2 Ra' F2 D2 F2 U2 Ra (11f*)
(8b+25b²+5f)
F2 L2 Ua B2 L2 Ua' B2 R2 (10f*)
(8b+25b²+5i+5f)
F2 L2 Ua' F2 R2 Ua B2 R2 (10f*)
(13h+25b²+5f)
B2 D2 F2 Ra' F2 D2 F2 U2 Ra' (11f*)
(13h+25b²+5i+5f)
F2 D2 B2 Ra' F2 U2 F2 U2 Ra' (11f*)

26. Symmetry m4, <(1234)>, 4.

This is a slightly unusual group, and it cannot be described with the notation used so far. It is of order 4, generated by the reflection/rotation I've denoted by m4 which is the combination of r4 with m_c. All previous symmetry groups that contained this element also contained r4 and m_c separately. The patterns in this group include those of sections 5, and 13.

The edges and corners all fall into orbits of 4 pieces; the tetrads of corners {URF,DLF,ULB,DRB} and {DRF,ULF,DLB,URB}, the edges {UF,DL,UB,DR}, {UF,DB,UR,DF}, and the middle layer edges. Any of these orbits can move around in a 4-cycle around the U/D axis, as long as this odd permutation is compensated for by a another 4-cycle.

4-cycle of corners, 4-cycle of edges (URF,DLF,ULB,DRB)(RF,LF,LB,RB)
R2 U2 B U2 B' U2 F U2 F' D2 B D2 F' R2 F
Two 4-cycles of edges type 4; 2H + 4K (UF,DL,UB,DR)(RF,FL,LB,BR)
D2 B L F' U2 R Us F' U2 L B' R' Ua' (15f*,18q*)

By using these in a different orientation, the other U/D edge/corner orbits can be cycled around as well. The middle layer can be cycled twice on its own by 5i.

The U/D edge orbits can be swapped by 5f, and the corner orbits by 5b. The U/D edges can be flipped by 11e, the middle edges by 5d. Each corner orbit can be twisted:

4-twist of corners, type 5 {URF+,DLF-,ULB+,DRB-}
F2 U Ra' D Ra U' F2 R Ua L' Ua' R' (18q*)

These patterns form a group of order 442,368 (= 2¹⁴3³).

Let's first consider the patterns that use adjacent colours. The corners either do not move at all, or move by 13a. This leaves very few nice patterns with exactly this symmetry:

2H + 4K (26b+5f)
F2 Rs' D2 B Ra D Rs' Us R' Fa U' (16f*)
2H, 4U (13a+26b)
L2 Fs' D2 L' Ua' F' Rs' Fs' R Fa' U (16f*)
2H, 4U (13a+26b+5f)
B R F' D2 L Us B' D2 L F' R' (12f*)

Now for patterns using opposing colours. The corner orbits cannot be swapped, so they both move by some multiple of 26a. The only essentially different combinations are then 5a, 10b, 12a', 13h, 26a, 26a+12a'. The r2_e symmetry of 10b (or the identity) cannot be broken by edge movements, and neither can the m_f symmetry of 5a.

(13h+10d)
Fa' (D2 L2)² Fa R2 U2 R2 (11f*)
(R2 F2)² D2 R2 U2 L2 B2 R2 B2 D2 L2
(13h+10d+5f)
Fa D2 R2 U2 R2 Fa L2 D2 R2 (11f*)
(F2 R2)² D2 B2 U2 F2 R2 F2 L2 U2 F2
(13h+10d+5i)
Fa' D2 R2 U2 R2 Fa R2 U2 R2 (11f*)
(R2 F2)² U2 L2 D2 L2 B2 R2 B2 D2 L2
(13h+10d+5f+5i)
Fa D2 L2 D2 L2 Fa L2 D2 R2 (11f*)
(F2 R2)² U2 F2 D2 F2 R2 F2 L2 U2 F2
(12a+10d)
D2 L2 F' D2 Rs2 U2 B' R2 (9f*)
(U2 F2)² R2 U2 R2 B2 D2 L2 B2 R2 F2 R2
(12a+10d+5f)
B2 R2 Ua B2 R2 Ua B2 R2 (10f*)
R2 (U2 F2)³ L2 D2 F2 R2 B2
(12a+10d+5i)
Ra Fa D2 Fa' U2 Ra' (10f*)
(U2 F2)² R2 D2 L2 F2 U2 R2 F2 R2 B2 L2
(12a+10d+5f+5i)
B2 R2 Ua' F2 L2 Ua' B2 R2 (10f*)
F2 (R2 D2)³ B2 U2 L2 F2 R2
(26a+10d)
R2 U F2 R2 U2 R2 D' F2 D' B2 R2 U2 L2 U' R2 U'
(26a+10d')
U L2 F2 D2 F2 U Rs2 D' Rs2 U' F2 U2 R2 U'
(26a+10d+10d')
B2 D' L2 B2 R2 U' L2 D' R2 U2 B2 U' F2 U2 R2 U'
(26a+5f)
B2 R2 F2 L2 F2 R2 D' L2 F2 U2 F2 D F2 U2 R2 U'
(26a+10d+5f)
L2 D' F2 R2 U2 F2 L2 D' B2 D L2 D2 F2 U' R2 U'
(26a+10d'+5f)
U R2 F2 U' R2 D L2 U B2 L2 D' L2 F2 U' R2 U'
(26a+10d+10d'+5f)
U L2 B2 D' F2 R2 U' B2 D' R2 B2 U R2 U' R2 U'
(26a+5i)
F2 L2 D' B2 U' B2 U L2 D R2 F2 U L2 U' R2 U'
(26a+10d+5i)
R2 D R2 B2 D2 F2 D' L2 U' B2 R2 U2 L2 U' R2 U'
(26a+10d'+5i)
D' L2 F2 D2 B2 D' Fs2 D' Fs2 U' F2 U2 R2 U'
(26a+10d+10d'+5i)
R2 U' L2 F2 D L2 U' B2 D' L2 F2 U' B2 U R2 U'
(26a+5f+5i)
D F2 D' F2 U F2 L2 U' F2 D' L2 B2 U' F2 R2 U'
(26a+10d+5f+5i)
L2 D' F2 D2 B2 R2 U L2 U' R2 U2 L2 F2 U' R2 U'
(26a+10d'+5f+5i)
U R2 F2 U' R2 U B2 U R2 B2 U' L2 F2 U' R2 U'
(26a+10d+10d'+5f+5i)
D F2 L2 D' R2 B2 D' B2 D' R2 B2 U R2 U' R2 U'
(26a+12a')
R2 D F2 R2 B2 U F2 U R2 U2 F2 D' F2 U2 R2 U'
(26a+12a'+10d)
D' L2 F2 D' R2 U L2 U L2 B2 U' B2 R2 U' R2 U'
(26a+12a'+10d')
D' R2 F2 U2 B2 U Fs2 U' Rs2 U' F2 U2 R2 U'
(26a+12a'+10d+10d')
D' R2 B2 D' R2 F2 D' L2 U' F2 L2 D F2 U' R2 U'
(26a+12a'+5f)
D B2 U B2 U' L2 B2 D L2 D L2 B2 D F2 R2 U'
(26a+12a'+10d+5f)
D L2 D Fs2 D R2 D' F2 U L2 F2 R2 U' R2 U'
(26a+12a'+10d'+5f)
D' F2 U R2 Fs2 D' R2 U' L2 U' B2 L2 U' R2 U'
(26a+12a'+10d+10d'+5f)
D L2 U Fs2 U L2 U' F2 U F2 L2 B2 U' R2 U'
(26a+12a'+5i)
R2 D F2 R2 B2 D R2 U B2 D2 L2 U' F2 U2 R2 U'
(26a+12a'+10d+5i)
U L2 B2 D2 F2 U' Fs2 D' Fs2 U' F2 U2 R2 U'
(26a+12a'+10d'+5i)
U L2 B2 D2 F2 U Rs2 D Rs2 U' F2 U2 R2 U'
(26a+12a'+10d+10d'+5i)
B2 D F2 L2 B2 U L2 D' R2 D2 F2 U F2 U2 R2 U'
(26a+12a'+5f+5i)
D' F2 D' F2 U F2 L2 U' F2 D' L2 B2 U' F2 R2 U
(26a+12a'+10d+5f+5i)
U B2 D Rs2 D F2 U' F2 U L2 F2 R2 U' R2 U'
(26a+12a'+10d'+5f+5i)
U R2 F2 U2 B2 D' Fs2 U' Rs2 D' B2 U2 R2 U'
(26a+12a'+10d+10d'+5f+5i)
U B2 U F2 L2 B2 U' L2 U L2 U' Rs2 U' R2 U'

27. Symmetry r3, <(123)>, 3.

Hold the cube so the r3 symmetry is around the UFR-BDL axis.

The corners occur in 4 orbits: {UFR}, {BDL}, {UFL, UBR, DFR}, {UBL, DFL, DBR}. As usual, the single corner orbits cannot be swapped, and can only be twisted in opposite directions by 15a. The other orbits can be cycled and twisted separately:

3-cycle of corners (UFL,FRD,RUB)
U R' B2 R F R' B2 R F' U' (10f*)
3-cycle of corners (UBL,FLD,RDB)
U B R' F2 R B' R' F2 R U' (10f*)
3-twist of corners {UFL+,FRD+,RUB+}
D' F2 U' R2 Ua B' D' F D' F D' B D' (14f*)
3-twist of corners {UBL+,FLD+,RDB+}
U' L2 D' B2 Ua R' U' L U' L U' R U' (14f*)

The three-corner orbits can be swapped only if edges also move, for example by using 7a.

The edges occur only in orbits of 3: {UF,FR,RU}, {DB,BL,LD}, {UL,FD,RB}, {UB,FL,RD}. These can be cycled separately:

3-cycle of edges [p196] (UR,FU,RF)
U R' Fs2 L F L' Fs2 R F' U' (12f*)
3-cycle of edges (FL,RD,UB)
U Fs D' F2 D Fs' R' F2 R U' (12f*)
3-cycle of edges (UL,FD,RB)
R2 D B2 F' U L U2 L' F B2 R' D' (12f*)
3-cycle of edges (DL,BD,LB)
D L F' Us2 B R' B' Us2 F D' (12f*)

The edge orbits can also be swapped. As it must be an even permutation, we must either have two orbit swaps (e.g. 1a, 3b), or a 3-cycle of orbits:

Three 3-cycles of edges (UF,DB,UB)(FR,BL,FL)(RU,LD,RD)
D B2 R2 U' Fs2 D R2 Us' R2 U' (12f*)
(R2 B2)² (U2 R2)² F2 U2 B2 D2
Three 3-cycles of edges (UF,UB,DF)(FR,FL,BR)(RU,FD,LU)
(R2 F2)² (U2 R2)² (F2 U2)² (12f*)

The edge orbits can only be flipped in pairs, by 7b, 7h and the following:

6-flip of edges {UF+,FR+,RU+,UB+,FL+,RD+}.
D2 L' F' R2 F2 U' B2 L2 R D F U2 F2 R B2 U' (16f*)

These patterns form a group of order 3,779,136 (= 2³3⁹4!).

Lets first consider all patterns that use opposing colours. The corners can move by 27a+27c' and/or 27b+27d. The edges move by 1a, 3b, or 27i, 27j. This leaves only the following:

(27a+27c')
F (R2 F')² U2 (F R2)² F' U2 (12f*)
(27a+27c'+27i)
D2 F2 L2 B2 R2 D' R2 D B2 D2 R2 U R2 U (14f*)
(27b+27d+27i)
B2 L2 F2 U L2 U B2 D2 R2 U F2 U' L2 U2 (14f*)
(27a'+27c+27i)
F2 L2 B2 L2 D U2 L2 D' B2 D2 L2 U' L2 U' (14f*)
(27b'+27d'+27i)
F2 D' L2 B2 U' R2 D2 B2 D R2 U2 B2 R2 U' (14f*)
(27a+27c'+27j)
R2 F2 R2 F2 U R2 U' R2 U2 F2 U' F2 U (13f*)
(27b+27d+27j)
R2 B2 D' L2 U R2 D2 R2 B2 D' R2 U R2 (13f*)
(27a'+27c+27j)
U' B2 D L2 U' F2 U2 B2 D B2 U' B2 U' (13f*)
(27b'+27d'+27j)
U F2 U F2 U2 R2 U R2 U' R2 F2 R2 F2 U2 (14f*)
(27a+27c'+1a)
D R2 D' B2 R2 D2 B2 R2 U' L2 U' R2 B2 U2 (14f*)
(27a+27c'+3b)
U B2 D' F2 R2 U2 F2 R2 U' F2 D' R2 F2 U2 (14f*)
(27a'+27c+3b)
U2 F2 R2 D F2 U R2 F2 U2 R2 F2 D B2 U' (14f*)
(27a+27c'+27b+27d+27i)
B2 Ua' R2 B2 U2 F2 Us' (9f*)
D2 R2 D2 B2 D2 R2 F2 R2 D2 R2
(27a'+27c+27b'+27d'+27i)
B2 D2 F2 Ra B2 D2 Ra' D2 B2 D2 (12f*)
B2 D2 F2 R2 U2 L2 D2 R2 F2 R2 F2 U2
(27a+27c'+27b+27d+27j)
F2 Ua R2 B2 U2 B2 Us (9f*)
U2 R2 U2 F2 U2 R2 F2 R2 U2 R2
(27a'+27c+27b'+27d'+27j)
F2 U2 R2 U2 R2 F2 R2 F2 R2 U2 F2 U2 (12f*)
(27a+27c'+27b'+27d'+27i)
L2 B2 R2 U L2 U' L2 F2 D2 F2 U B2 U' (13f*)
(27a'+27c+27b+27d+27i)
Fs2 Rs2 D L2 D' L2 D2 F2 U' R2 U (13f*)
(27a+27c'+27b'+27d'+27j)
F2 Rs2 F2 U' B2 R2 U L2 U2 F2 U B2 R2 U' (15f*)
6pl [p176] (27a'+27c+27b+27d+27j)
D R2 D' R2 U2 F2 U' R2 U (9f*)
(27a+27c'+27b'+27d'+3b)
U' F2 L2 U' B2 D2 R2 D U2 F2 R2 U' (12f*)

There are quite many patterns with adjacent colours, cycled around the axis of symmetry. Note that by doing the inverse of any sequence here, you get the same pattern with the colours cycled in the opposite direction. This is often the mirror image of the original pattern, but not always. I will not list those separately. First those which combine 27a, 27b, 27e-h and 15a.

(27e+27f)
Us' L B D2 R' Us2 L U2 B' L' Us (10f*)
(27a+27e)
U R' F2 L F L' F2 R F' U' (10f*)
(27a+27f)
D B' L2 B L U' L2 U L' D' (10f*)
(27a+27e+27f)
U (L' B')² U R2 (U' B L U')² (14f*)
(27a+27f+27g)
D2 B' L U B U' B L U' L U B L' D2 (14f*)
(27a+27e+27f+27g)
L' F' L2 U' F' D2 B Rs' Ua' B' R D' (14f*)
(27a+27h)
U F L B' L' D' L D B2 D' B2 L' B D F' U' (16f*)
(27a+27e+27h)
U' B R2 B' U F' U Fs' D' B' D' L' D' F2 U' (16f*)
(27a+27f+27h)
F L U' L U L F' R' D L2 D' B Rs D' (15f*)
(27a+27e+27f+27h)
D2 U' L' F2 L B' F2 R' B2 L D' Fs L F' D2 U (17f*)
(27a+27f+27g+27h)
D L B2 R' F' L Us' F' U2 R B2 U B' Ra' D' (17f*)
[p20] (27a+27e+27f+27g+27h)
L' F' L2 D' Fs' D2 Rs' Us' L B2 L' F' R D' (17f*)
(27a+15a)
D' B R' F' R B' D2 R D2 R2 F R2 D2 R' D' (15f*)
(27a+27e+15a)
U' L B' R' F2 R' F' R2 Fs' L' U' (12f*)
(27a+27f+15a)
D F2 D' B L' D2 L D F D2 F D' B' (13f*)
(27a+27e+27f+15a)
L2 D L2 U' L B D F' L D2 R' B' U' L2 (14f*)
(27a+27f+27g+15a)
R2 B' U' F' L B D2 F2 L D F' U' R2 D2 L U2 (16f*)
(27a+27e+27f+27g+15a)
L D' B2 U F' Rs' Ua F' U2 B2 F' L' D' (15f*)
(27a+27h+15a)
F D F' U F L' D F D2 L U2 R2 B R2 U2 F' U' (17f*)
(27a+27e+27h+15a)
U' R2 D B U R' F' R' F' U L D F2 U' (14f*)
(27a+27f+27h+15a)
R2 D F' U' F2 U' L2 F' L D' R2 B' D L2 D' (15f*)
(27a+27e+27f+27h+15a)
R' F' L F2 R' Us' Fs' L F R2 F' U (14f*)
(27a+27f+27g+27h+15a)
L2 U F U F R2 D B R D2 U' B D2 R F2 L D' (17f*)
(27a+27e+27f+27g+27h+15a)
L U' R2 D L D' R2 D L' Fs' Rs' U' (14f*)
(27e+27f+27g)
U2 L2 D' B' Rs' B2 L' D' L U L2 F L' U2 (15f*)
(27e+27f+27h)
U2 L2 D' F' L' R F2 R' D' R U R2 B L2 R U2 (16f*)
(27e+15a)
B2 U' R2 B2 D B D' B R' U R' B2 (12f*)
(27f+15a)
R D' U L' U2 L D L2 U' F' U2 F R' U L2 U' (16f*)
(27e+27f+15a)
F U B' L' B2 R B2 L2 F U F2 U' L' Fs' U' (16f*)
(27e+27f+27g+15a)
L U' L2 F2 L2 F D2 B' F2 Ra' Ua L2 B' D' (16f*)
(27e+27f+27h+15a)
R2 D B R' F' L F U' F' R' D F2 U F U' F U2 (17f*)
(27a+27b+27e)
U B R' F2 R B' Rs' F L' F2 R F' U' (14f*)
[p87] (27a+27b+27f)
U R2 U' F' U2 F2 U2 F R F2 R' U R2 U' (14f*)
(27a+27b+27e+27f)
F D' F2 D U L D F' D' Ra' F2 R U' (14f*)
(27a+27b+27e+27f+27g)
R2 D' B2 R2 U R U' L Us' Fs' R' B L' U2 (16f*)
(27a+27b+27e+27f+27h)
R' B R F2 L F2 R' F D2 R' Ua' B R2 F' U (16f*)
(27a+27b+27e+15a)
U R D2 R' F Us' R2 Us' R' F' D2 F U (15f*)
(27a+27b+27f+15a)
R B2 D' B D R2 B R D' F' D2 R' F' R F2 D' (16f*)
(27a+27b+27e+27f+15a)
R2 B2 U' L Us' F' U L2 B D' Rs' B U (15f*)
(27a+27b+27e+27f+27g+15a)
Us' B L D2 L' F' R' B' D2 B L Us' (14f*)
(27a+27b+27e+27f+27h+15a)
Us' R U' Fs' L U' R' Us' F (12f*)

Now those patterns which use 15r=15d'+7b combined with 27a, 27b, 27f, 27g and 15a.

(15r+27f)
F U L U Rs' B F2 D R B R' B' D2 L' D' (16f*)
(15r+27f+15a)
B' U' F2 R' F' U B R2 Fs2 L' U' B' L2 U' F R U2 (18f*)
(15r+27a)
R2 B' Ra' B' U' Fa' L' F U L2 F U2 F' R' U' (17f*)
(15r+27a+27f)
B L U' B2 L' D B' L2 B L2 F' D' B2 D2 U R' D2 U' (18f*)
(15r+27a+27g)
R' B' R2 D Fs Rs D2 F U' F' L B L R2 F' U2 (18f*)
(15r+27a+27f+27g)
L' D2 F D R2 B' R' U2 R' D2 L2 B F2 R' F' D F U' (18f*)
(15r+27a+15a)
R' B R F R' B' D F' R2 U2 B L U2 R' U L' U' (17f*)
(15r+27a+27f+15a)
U R D2 F' D2 R' Ua' L' Us' R F2 R' F' L D' (17f*)
(15r+27a+27g+15a)
D' B R' F' R2 F Us' B' Ua' F' D2 R' D2 F U (17f*)
(15r+27a+27f+27g+15a)
R B L B' R' Ua' F2 D2 F U B F2 R2 F2 R U' (17f*)
(15r+27a+27b+27f)
B R Fa' D2 U F U B2 U2 B' D' B' F2 Ua L' U' (18f*)
(15r+27a+27b+27f+15a)
B R F U L2 Us' B' U Fs' R2 B' D' L' U' (16f*)

Finally, there are a few exceptional patterns that have other colour combinations which are only possible by moving just edges.

R2 Us R F R2 D2 R2 F R2 B2 D2 R' F2 Ua (16f*)
F2 U2 R2 D R2 B2 L F' D U2 L2 D B R U' F2 U2 (17f*)
Rs2 U' B' L' R2 B2 L' Fs2 L D2 R' B U Rs2 (17f*)
D2 U B' U2 R' Us' B' F2 U2 Fs2 R D' U2 (15f*)
F' L' D B L2 Fs D' L' B' U' L F R' U R U' (17f*)
R2 U B2 F Rs2 F D Rs D' U2 Rs B' L' U' R2 (18f*)

28. Symmetry m_f, <(13)(24)>, 2.

Hold the cube so that the plane of reflection goes through the middle layer.

The U/D pieces form 8 orbits, consisting of one U piece together with the D piece directly below it. The middle layer edges form orbits by themselves. The two pieces in each U/D orbit can swap, and any two similar orbits can be exchanged, provided the whole cube has an even permutation. Any corner orbit can be twisted separately (pieces in opposite directions), any U/D edge orbit can be flipped, and any two middle layer orbits can be flipped.

These patterns form a group of order 18,345,885,696 (= 4!³ 3⁴ 2¹⁴ ). Even if we limit ourselves to patterns with at most 2 colours per face there are 2277 patterns, so only some representative examples will be listed in this section.

The corner orbits will not be twisted (that would give too many colours on a face), but this still leaves about 15 other distinct ways the corners could move. First suppose the corners do not move at all. The following list contains all patterns which have top and bottom faces with only one colour:

Edge 3-cycle (FL,BR,FR)
F2 Um F2 Um' (8f*)
3H pattern [p173]
U2 R2 B2 D' Fs2 U F2 R2 U2 (10f*)
2K, 2A
Rs2 U2 L2 Us F2 Ua' (9f*)
2K, 2A
Rs2 Ua F2 Us' L2 U2 (9f*)
4K
D B2 L2 Fs2 Us' F2 R2 U' (10f*)
4H
Rs2 U2 Fs2 U Fs2 U2 Rs2 U' (12f*)

By swapping various U and D edges in the patterns above, quite a few more can be produced. Here is a selection:

4H
U2 B2 R2 D' B2 F2 U L2 F2 U2 (10f*)
5H
Fs2 D F2 Ra' U2 Ra' F2 R2 U' (12f*)
5H
Rs2 D L2 F2 Ra' U2 Ra F2 U' (12f*)
3H, 2X
U2 F2 L2 U Fs2 D' L2 F2 Rs2 U2 (12f*)
6H
U2 Fs2 D' F2 Rs2 F2 Rs2 U' (11f*)
6H
Fs2 D F2 Rs2 F2 Rs2 U' (10f*)
6H
R2 B2 R2 U' L2 D B2 R2 B2 D F2 U' (12f*)
4A, 2H
R2 Us B2 L2 Fs2 R2 Us' (10f*)
4A, 2H
F2 L2 Us Rs2 F2 R2 Us' (10f*)
4A, 2X
D2 L2 Us' B2 Rs2 Ua' (9f*)
[p37]
F2 L2 F2 L2 F2 Us2 R2 Us2 (10f*)
2K, 2H, 2A
D2 F2 R2 Us' F2 R2 Ua' (9f*)
2K, 2X, 2A
D2 F2 U2 R2 D2 F2 U2 R2 (8f*)
R2 F2 R2 F2 Us' F2 R2 Us' (10f*)
4K, 2H
L2 Us' Rs2 F2 Us' (8f*)
4K, 2H
F2 R2 Us' F2 R2 Us' (8f*)
4K, 2X
L2 Us' B2 Rs2 Us' (8f*)
[p39]
L2 Fa U2 F2 U2 Rs2 Fa' (10f*)
B2 R2 F2 R2 U2 Fs2 U2 F2 R2 (10f*)
F2 L2 F2 R2 U2 Rs2 U2 F2 R2 (10f*)
[p28]
F2 L2 F2 L2 Us2 F2 R2 (8f*)
[p43]
R2 Fa' U2 R2 Fa' R2 F2 U2 (10f*)
U2 Fs2 U2 (L2 F2)² R2 B2 (10f*)

The following have an odd permutation in the middle layer, and to compensate the UF and DF edges are also swapped. The rest of the U and D facelets are not changed. Again, only a selection of the possible patterns is listed here.

Double edge swap (FU,FD)(FR,BR) [p35]
R F2 R2 F2 R2 F2 R (7f*)
D2 F2 L2 U2 R2 B2 U2 L2
Double edge swap (FU,FD)(FR,BL)
R2 F' Us2 B2 Us2 F' R2 (9f*)
Double edge swap (FU,FD)(FR,LB)
R B' R2 D L2 F2 R2 U L2 B' R' (11f*)
6A
D2 B2 L2 U2 L2 D2 R2 U2 F2 U2 (10f*)
4H, 2A
D' B L B' L2 R F U' L D F' R' B' F2 (14f*)
2H, 4A
L2 Ua' L2 B2 D2 F2 R2 Ua' (10f*)
2K, 2A
U2 L2 D2 Ra' F2 Ra' (8f*)
2K, 4A
R2 B' Rs2 Fa' Us2 F' R2 (10f*)
B2 L2 U2 F2 L2 D2 R2 F2 U2 F2 (10f*)
4K, 2A
R2 B2 F' L2 Us2 R2 B' L2 (9f*)
2K, 4A
L2 Ua' B2 Ua' Ra' F2 Rs' (11f*)
2K, 4A
R2 B' F2 D2 Rs2 U2 F' Us2 R2 (11f*)
Fs2 U2 F2 L2 U2 L2 D2 R2 U2 F2 U2
4K, 2A
L2 Us2 F' Rs2 F2 Rs2 F' R2 (11f*)
2X, 4A
Ra Us2 B2 Ra' D2 R2 U2 (10f*)
4K, 2A
B2 L2 U F2 L2 B2 Rs2 U R' F D Rs2 U' B' R' (17f*)
4X, 2A
Rs2 D2 Fs' Us2 L' Ua F Rs Fs R' Fa' D' (19)
4X, 2A
D2 Fs' L' Ua B Rs Fs' L2 R Fa' D' (16f*)

By swapping various U and D edges in the patterns above, many more patterns can be produced, such as the following:

2K, 4A
Fs2 L2 F' Us2 F2 Us2 F' R2 (11f*)
2K, 4A
U2 Fs2 D' L B U Fs2 D' B' R' U' (13f*)
2K, 4A
U2 F2 U2 Ra' F2 U2 Ra' U2 R2 U2 (12f*)
U2 F2 R2 D2 R2 D2 R2 U2 F2 L2 R2 U2 (12f*)
2K, 4H
B' R2 Fs2 L' F U' R' D F' L' R2 B R' B' U (16f*)
4K [p45]
L2 Ua L2 F2 D2 F2 R2 Ua' (10f*)
R2 U2 B2 (R2 D2)² U2 B2 D2 (10f*)
4K, 2H
B2 L2 B L2 Us2 R2 F' L2 F2 (10f*)
6K
R2 F L2 Us2 R2 B' R2 Fs2 (10f*)
Fs2 U2 B2 L2 U2 R2 U2 R2 U2 F2 U2
6K
F2 D2 R2 U2 F2 Ra' B2 Ra' (10f*)
B2 R2 D2 F2 R2 U2 R2 F2 U2 F2 (10f*)
4K, 2X[p46]
L2 D2 B2 L2 U2 L2 B2 U2 Fs2 (10f*)
4K, 2A
Rs2 B D' L' F2 U' F2 L' D B' D L2 B R2 U (16f*)
6K
U' F2 Rs2 F2 R2 L B' D' Fs2 U B R U (15f*)
2K, 4X
Rs D2 B R2 Us F2 R' D2 R2 Fs U' Rs2 U' (17f*)
2K, 4X [p138]
Rs D2 F R2 Us' B2 R Fs' D2 L2 U Fs2 U' (17f*)

There is a huge number of pretty patterns in this group in which the corners also move. An easy way to generate many of them is to take any pattern above which uses only opposing colours, and do a half turn of a slice or side face (or several). First I'll list just a few in which a slice is given a half turn.

2O, 2K, 2t [p29]
L2 U2 F2 R2 D2 L2 F2 U2 (8f*)
[p47]
R2 F2 R2 Us R2 F2 R2 Us' (10f*)
(L2 F2)² Us2 L2 Us2 F2 (10f*)
2+, 2K, 2U [p89]
Fs2 L2 U2 B2 R2 U2 R2 F2 U2 (10f*)
2+, 2H, 2v [p90]
Fa' D2 L2 Fa R2 F2 U2 R2 (10f*)
B2 R2 U2 Fs2 U2 B2 R2 B2 L2 (10f*)
2O, 2v [p41]
R2 Fa U2 B2 D2 Fa' (8f*)
U2 Fs2 D2 R2 B2 L2 B2 R2 F2
2+, 2v [p40]
F2 Us2 L2 B2 R2 B2 L2 Us2 (10f*)
2I, 2v
Fa U2 F2 U2 Rs2 Fa' R2 (10f*)
F2 R2 B2 L2 U2 Fs2 U2 F2 R2 (10f*)
2U, 2t
R' B2 R2 B2 L2 F2 R' (7f*)
4U
D2 L2 U2 Ra F2 Ra' (8f*)
4U, 2X
L2 U2 F2 L2 U2 Fs2 R2 F2 U2 (10f*)
Ra' F2 Ra U2 Fs2 R2 U2 (10f*)
4t
L2 U2 Fa U2 L2 U2 Fa' U2 (10f*)
4t, 2A
Fs2 R2 F' Us2 F2 Us2 F' R2 (11f*)
U2 F2 L2 D2 L2 D2 R2 U2 F2 Rs2 U2
2U, 2t
R2 Ua L2 F2 D2 F2 R2 Ua' (10f*)
4U, 2A
Us2 B2 R D2 Fs2 U2 L' B2 (10f*)
4U, 2A
R2 U2 B2 U2 R2 Fa L2 Fa' (10f*)
4t
L2 U2 Fa U2 L2 U2 Fa' U2 (10f*)
2t, 2U
R2 Ua L2 F2 D2 F2 R2 Ua' (10f*)
4U [p88]
D F2 Us' R2 U' (6f*)
D2 L2 D2 B2 D2 L2 U2 F2
2t, 2U
Rs2 D2 L2 Us F2 Ua' (9f*)
2t, 2U
Rs2 Ua' F2 Us' L2 U2 (9f*)
4t
D' B2 L2 Fs2 Us' F2 R2 U (10f*)

Here are a few that have half turns of the sides.

6T [p97]
R2 F2 Us' F2 R2 Us' (8f*)
R2 F2 U2 F2 D2 R2 (U2 B2)²
4T, 2v [p104]
F2 D2 F2 L2 U2 L2 F2 U2 F2 (9f*)
6c [p110]
L2 Us R2 B2 Us' F2 (8f*)
L2 B2 U2 F2 U2 L2 (U2 F2)²

Patterns with a 4-cycle of side colours such as 28af can be combined with a Us slice move. Other unusual edge patterns can also sometimes be combined with a matching corner permutation. This gives patterns like for example these:

4O, 2A
R2 D' B R B' L R2 F D' R U F' L' B' (14f*)
4O, 2K[p144]
D' R F R' F2 B L D' F U L' B Rs2 F2 R (16f*)
6t [p147]
U' F' L' U' Fs2 D R F' B2 D2 R2 Us2 R2 U' (16f*)
U2 F2 D' R2 U F2 U' R2 U F2 D' R2 U2 (13f*)
B2 L' U B2 R D' R U F2 R' D' R' (12f*)
R2 F2 U2 L2 D B2 U' L2 U B2 D' B2 U F2 U2 (15f*)
D B2 D' L2 D R2 B2 U F2 U' L2 U' R2 F2 U' (15f)

29. Symmetry m_e, <(13)>, 2.

Hold the cube so that the plane of reflection is perpendicular to the FR-BL axis.

The corner orbits are {URF, ULB} and {DFR, DBL}, and the other four corners form an orbit each. The edges form seven orbits, viz. {UR,UB}, {UL,UF}, {DR,DB}, {DL,DF}, and the middle layer orbits {FR, BL} and {FL} and {BR}. As usual, any two similar orbits can be exchanged. The two larger corner orbits and all the edge orbits can be oriented, and two pieces in an orbit can be swapped.

These patterns form a group of order 424,673,280 (= 5! 4! 3² 2¹⁴ ).

There are 344 patterns with at most two colours per face, so I will not list them all. Here are some of those that only move edges.

R2 U' Rs2 D R B' D' Rs2 U F R (13f*)
L2 F2 D' Fs2 U R' B D Fs2 U' B' L F2 L2 (16f*)
D F2 D2 U Fs' L B2 L2 Us' F Rs (14f*)
Fs2 L' R F R2 B2 Us' R Fs D B2 D' (15f*)
L2 U R2 U Rs2 D' L2 U' L2 (10f*)
R2 U R2 D Rs2 U' L2 U' R2 (10f*)
B2 L' Fs2 R D2 R B U' B' D B F' L F' L' U B' (18f)
Rs2 D2 B2 R' F2 U' L2 B2 L R2 B' L' F2 U' L' B' U2 (18f)
R2 F' L2 U' F R2 D2 R U' R' D' B' D' U2 R2 B2 U' (17f)

There are many patterns, but hardly any interesting ones.

U2 B2 R2 U R U' R B2 L' D L U2 (12f*)
[p54]
Fs' Ua L2 Us Fa' R2 (10f*)
B2 L2 D2 R2 D2 F2 R2 B2 U2 F2 U2 L2

30. Symmetry m_c, <i>, 2.

The pieces fall into 10 orbits, each comprising two diametrically opposite pieces of the cube. Each pair in an orbit can be swapped, similar orbits can be exchanged, and all orbits can be oriented.

These patterns form a group of order 45,864,714,240 (= 6! 4! 3⁴ 2¹⁵ ).

31. Symmetry r2_f, <(13)(24)>, 2.

Hold the cube so that the r2 axis is centred on the U/D faces.

The corners fall into four orbits, {UFR,UBL}, {UFL,UBR}, {DFR,DBL}, {DFL,DBR}, and the edges into the 6 orbits {UR,UL}, {UF,UB}, {DR,DL}, {DF,DB}, and {FR,BL}, {FL,BR}.

These patterns form a group of order 15,288,238,080 (= 6! 4! 3³ 2¹⁵ ).

[p69]
D' F U2 Ra' F2 U R2 D L2 U Fs' R' U' (15f*)
[p79]
L U2 F2 R' Fs2 L' R2 B2 U2 R' (11f*)
F2 Rs2 D2 F2 L2 Fs2 D2 F2 U2 L2
[p91]
Fa' U2 L2 D2 R2 Fa R2 U2 R2 U2 (12f*)
L2 D2 B2 U2 B2 L2 F2 L2 (U2 F2)² R2 F2
[p102]
B2 D2 L2 B2 D2 L2 F2 D2 R2 (9f*)
[p105]
D2 F2 L2 F2 Ra' B2 Rs' F2 (10f*)
R2 F2 R2 D2 B2 L2 B2 R2 B2 U2 F2
[p107]
R2 D2 F2 R2 U2 B2 R2 U2 F2 (9f*)
[p127]
F2 Ua L2 Ua' F2 U2 (8f*)
D2 R2 U2 F2 D2 L2 F2 U2 B2 U2
[p126]
D R2 U2 F2 D' Fs2 U2 Rs2 U' F2 U2 R2 U' (15f*)
B2 U R2 U2 B2 D2 R2 B2 D2 F2 D' Fa' U2 Fs'

32. Symmetry r2_e, <(24)>, 2.

Hold the cube so that the r2 axis is centred on the FR-BL edges.

The corners fall into four orbits, {UFR,DFR}, {UBL,DBL}, and {UFL,DBR}, {UBR,DFL}, and the edges into the 7 orbits {UF,DR}, {UR,DF}, {UB,DL}, {UL,DB}, and {FL,BR}, {FR}, {BL}.

These patterns form a group of order 2,548,039,680 (= 5! 4! 3³ 2¹⁵ ).