6 x order 3     ->  4 3-cycles of edges
                    (uf ur fr) (ul br df) (ub dr fl) (bl db dl)
6 x order 2     ->  (uf db) (ul dr) (ub df) (ur dl) (fr bl) (fl br)
                    6 2-cycles of edges
6 x order 6     ->  (uf bl ur db fr dl) (ul ub br dr df fl)
                    2 6-cycles of edges

repeating
---------

EDGE/2p 1 (X)1 (1:8 3:4) 3 250                                      6x order 3
CORNER/3p EDGE/0p CENTER/2p 1 (X)1 (1:12 3:6) 3 517                 6x order 3
EDGE/2p CENTER/1o 1 (X2)1 (1:14 3:4) 3 250                          6x order 3
(F' U' D L 2F' 2L' 2F 2L)11                                         6x order 3
(2L B' 2L2 F 2D' 2F' 2D 2F Lv')4                                    6x order 3
(U' 2F 2L 2F' 2L D R' L)4                                           6x order 3
(U' F' B Fv L' R Lv F B' U' D F D' U Fv' Lv)4                       6x order 3
(U' F' B D' U L R' B' F L F' B Rv' Fv' )4                           6x order 3
(U' F' B D' U L R' B' F L F' B Lv Fv')4                             6x order 3
(R2 L3 D1 F2 R3 D3 F1 B3 U1 D3 F1 L1 D2 F3 R1 L2)2                  6x order 3
(32 moves)
(F Lv Fv' L' B F' D' B2 F2 U R L' U' R2 L2 U)2                      6x order 3
(28 moves)
(D' 2L2 U 2L' 2F' Lv2 B' 2L2 F 2L' 2F')2                            6x order 3
(20 slice moves)
(D' 2L2 U 2L' F' 2F' R2 L2 F 2L' 2F')2                              6x order 3
(22 slice moves)

EDGE/2p CENTER/1o 1 (X)1 (1:14 6:2) 6 250                           6x order 6
(F1 R1 L3 U2 F2 B2 U3 D3 F1 B3 L3 U1 R2 L2 U2 F2 B2 D1)7            6x order 6
(F1 R1 L3 U2 F2 B2 U3 D3 F1 B3 L3 U1 R2 L2 U2 F2 B2 D1)5            6x order 6
(90 moves)

6 X order 3 solutions

non-repeating
-------------

// Rubik's Compendium Method
// Earliest 6 X order 3 process

(R' L2 F2 B2' U2 R2 L2' F2 B2' D2 R' Uv)2
2L 2D' 2L' 2D

R2 L' D F2 R' D' F B' U D' F L D2 F' R L2
16 moves

2F U 2F2 U2 2D2 2L2 U L 2F2 L2 2D2 L 2F' 2D2  
2F D 2F2 2L2 D L' 2F2 2D2 L' U2 R2 2F U2 L2   
2F 2D2 2L2 U 2F2 U2 2L2 U L 2F2 L2 2D2 L 2F'
14 slice moves

2F U 2F2 1-2U2 2L2 U L 2F2 L2 2U2 L 2F' 2U2
13 block moves (here we used face + slice + slab)

((D' 2F2 U 2F2)3 Lv')2
2F 2D 2F' 2D'
mark's method for 6 x order 3

DeepCubeA's solution

F D F B L R' U U B D U F' U' B' U B B R B' D' B' D F' D' B R B' R' B' D B F 
32 moves

ASCII representation
--------------------


6 X order 3
             TOP
           0  2  0
           2  0  2
           0  2  0
    LEFT    FRONT    RIGHT      BACK
  4  1  4  2  5  2  5  0  5    3  4  3
  1  4  1  5  2  5  0  5  0    4  3  4
  4  1  4  2  5  2  5  0  5    3  4  3
            DOWN
           1  3  1
           3  1  3
           1  3  1

6 X order 2
             TOP
           0  1  0
           1  0  1
           0  1  0
    LEFT    FRONT    RIGHT      BACK
  4  5  4  2  3  2  5  4  5    3  2  3
  5  4  5  3  2  3  4  5  4    2  3  2
  4  5  4  2  3  2  5  4  5    3  2  3
            DOWN
           1  0  1
           0  1  0
           1  0  1