Classification of Groups on up to 7 Working Bells

The list of permutation groups below, useful for composition in parts and (hopefully) exhaustive, is classified according to:

  1. The number of working bells in the group; i.e. the number of bells which are not fixed in each part of a part composition (fixed bells can always be appended to increase the number of bells actually involved).
  2. The overall partition, if any, of the working bells; i.e. the segregation of the working bells into sets, bells only exchanging with those in the same set. Transitive groups have no partition.
  3. The order of the group, i.e. the number of permutations in it. In the list, the order is followed by "p" (all perms positive) or "m" (mixed, i.e. equal numbers of +ve and -ve perms).
  4. The signature of the permutation group. Each permutation element is classified according to the nature of the sub-group generated when its transposition (or transfigure) is repeated from rounds. Although this sub-group will technically always be cyclic, what is relevant here is the segregation of the working bells. For example 234165 generates the sub-group (123456, 234165, 341256, 412365) which in the general mathematical sense is a cyclic group of order 4; here we are concerned with the partition of the 6 bells of a perm into cycles of 4 and 2.

The signature of a group specifies the number of perms for each kind of partition. For instance,

                 0--------------0
                 | 4,1,1    N=6 |
                 | 3,3      N=8 |
                 | 2,2,2    N=6 |
                 | 2,2,1,1  N=3 |
                 0--------------0

The permutation group with this signature must be of order 24, since the frequencies total 23 and rounds is omitted in the analysis, as being a trivial (1,1,1,1,1,1) and always present. The orders of the cycles in each line total 6, the number of working bells. Six of the 24 elements have cycle sets (4,1,1) i.e. rotate 4 working bells and leave two unaltered; and so on.

In practice, the signature provides a way of identifying whether or not two permutation groups are equivalent in the change-ringing sense (i.e. one may be transformed into the other by choosing a suitable transposition and/or transfigure). It provides a way of identification which is amenable to being programmed as an algorithm. It also enables one to see at a glance whether "forbidden" kinds of transpositions, and the possibility of call shunts (relevant to particular methods), are present in a group being considered for part composition.

The signature has hidden relationships between its parameters. For instance, the sum of the number of symbols left unchanged in all the group perms bears a relation to the partitions of the group (Burnside p.191). In the example above, remembering that rounds is implied but omitted from the signature, the frequency of occurrences of the number 1 in the signature is totalled thus:

                 0--------------0
                 | 4,1,1    N=6 |     2 x 6 = 12
                 | 3,3      N=8 |
                 | 2,2,2    N=6 |
                 | 2,2,1,1  N=3 |     2 x 3 =  6
                 0--------------0
             Rounds (1,1,1,1,1,1)     6 x 1 =  6    Total = 24

For transitive groups, this total is equal to the order of the group. For groups with figures (over all the elements) in p partitions, it is p times the order. Hence the transitivity or number of partitions of the bells of a group can be deduced from its signature. For another relationship, see Burnside p.52 Corollary I.

It may be observed that N is odd only for entries involving only 2 or 1, and is either odd or zero if there are just two 2s.

The distinction between Primitive and Imprimitive transitive groups (Burnside p.191) was thought not to be useful in the classification. Of the transitive groups of order 6 listed below, only [6.02], [6.03] and [6.05] are primitive. In the others, sets of bells (2 sets of 3 or 3 sets of 2) remain together as they gyrate.

Permutation Groups on up to 7 Working Bells


2 working bells, transitive

0--------0
| 2  N=1 |     [2.01]  Order 2m
0--------0

The extent on 2 bells. Subgroups: none; Parent groups: 3.01 4.06 5.08 6.36 7.14; Normal supergroups: 4.06(4m) 5.06(12m) 5.08(6m) 6.17(48m) 6.18(24m) 6.20(16m) 6.21(8m) 6.22(8m) 6.34(8m) 6.36(4m) 7.08(240m) 7.10(120m) 7.11(40m) 7.13(20m) 7.14(10m) 7.34(24m) 7.37(12m) 7.38(12m);


3 working bells, transitive

0----------0
| 3    N=2 |     [3.01]   Order 6m
| 2,1  N=3 |
0----------0

The extent on 3 bells. Isomorphic with the group of rotations of the triangular prism. Subgroups: 2.01 3.02; Normal subgroup: 3.02; Parent groups: 4.01 5.06 6.29 7.36; Normal supergroups: 5.06(12m) 6.28(36m) 6.29(18m) 7.16(144m) 7.18(72m) 7.20(48m) 7.26(24m) 7.27(24m) 7.34(24m) 7.36(12m).


0--------0
| 3  N=2 |     [3.02]   Order 3p
0--------0

The cyclic group on 3 bells. Isomorphic with the group of rotations of an equilateral triangle. Subgroups: none; Parent groups: 3.01 4.02 5.07 5.08 6.31 7.39 7.40; Normal supergroups: 3.01(6m) 5.06(12m) 5.07(6p) 5.08(6m) 6.28(36m) 6.29(18m) 6.30(18p) 6.31(9p) 7.16(144m) 7.17(72m) 7.18(72m) 7.19(72p) 7.20(48m) 7.21(36p) 7.22(24m) 7.23(24p) 7.24(24m) 7.25(24m) 7.26(24m) 7.27(24m) 7.29(12m) 7.30(12p) 7.31(12p) 7.32(12m) 7.34(24m) 7.35(12p) 7.36(12m) 7.37(12m) 7.38(12m) 7.39(6p) 7.40(6m).


4 working bells, transitive

0------------0
| 4      N=6 |     [4.01]   Order 24m
| 3,1    N=8 |
| 2,2    N=3 |
| 2,1,1  N=6 |
0------------0

The extent on 4 bells. Isomorphic with the rotation group of a cube, opposite pairs of vertices being labelled 1 to 4. Primary subgroups: 3.01 4.02 4.03; Normal subgroups: 4.02 4.04; Parent groups: 5.01 6.17 7.17; Normal supergroups: 6.17(48m) 7.16(144m) 7.17(72m).


0----------0
| 3,1  N=8 |     [4.02]   Order 12p
| 2,2  N=3 |
0----------0

1234 4321 1342 2431 1423 3241  
2143 3412 3124 4213 4132 2314  

All the +ve permutations of 1234. The alternating group. Isomorphic with the rotation group of the tetrahedron, vertices labelled 1 to 4. Primary subgroup: 4.04; Subgroup: 3.02; Normal subgroup: 4.04; Parent groups: 4.01 5.02 6.18 6.19 7.21; Normal supergroups: 4.01(24m) 6.17(48m) 6.18(24m) 6.19(24p) 7.16(144m) 7.17(72m) 7.18(72m) 7.19(72p) 7.21(36p).


0------------0
| 4      N=2 |     [4.03]   Order 8m
| 2,2    N=3 |
| 2,1,1  N=2 |
0------------0

1234  2341  3412  4123    4321  3214  2143  1432

The dihedral group on 4 bells; a 4-part cycle and its reverse. Isomorphic with the rotation group of a square prism, side faces labelled 1 to 4. Primary subgroups: 4.04 4.05 4.06; Normal subgroups: 4.04 4.05 4.06 4.07*; Parent groups: 4.01 6.20 7.22; Normal supergroups: 6.20(16m) 7.20(48m) 7.22(24m).

* [4.03] has two different kinds of (2,2) transpositions; 3412 above corresponds to a half-turn about the 4-fold axis of the prism, whereas 4321 or 2143 are on the reverse cycle and involve turning the prism over about a 2-fold axis through side faces. It is only the first kind which gives 4.07 a normal subgroup of 4.03. This distinction will crop up repeatedly in groups involving group 4.03, and the first kind will be termed opposite pairs or opposites as the pairs swapping are opposite in the 4-cycle of the group.


0----------0
| 2,2  N=3 |     [4.04]   Order 4p
0----------0

1234   2143   3412   4321

The `Pairs of pairs' group, isomorphic with the rotation group of a cuboid, a set of four equal parallel edges being labelled 1 to 4. Subgroup: 4.07; Normal subgroup: 4.07; Parent groups: 4.02 4.03 6.21 6.23 7.31 7.33; Normal supergroups: 4.01(24m) 4.02(12p) 4.03(8m) 6.17(48m) 6.18(24m) 6.19(24p) 6.20(16m) 6.21(8m) 6.23(8p) 7.16(144m) 7.17(72m) 7.18(72m) 7.19(72p) 7.20(48m) 7.21(36p) 7.22(24m) 7.23(24p) 7.27(24m) 7.28(24p) 7.31(12p) 7.33(12p).


0----------0
| 4    N=2 |     [4.05]   Order 4m
| 2,2  N=1 |
0----------0

1234   2341   3412   4123

The cyclic group on 4 bells. Isomorphic with the rotation group of a square, corners labelled 1 to 4. Subgroup: 4.07; Normal subgroup: 4.07; Parent groups: 4.03 5.03 6.22 6.25 7.29; Normal supergroups: 4.03(8m) 6.20(16m) 6.22(8m) 6.25(8m) 7.20(48m) 7.22(24m) 7.24(24m) 7.26(24m) 7.29(12m).


4 working bells, partitioned (2, 2)

0------------0
| 2,2    N=1 |     [4.06]   Order 4m
| 2,1,1  N=2 |
0------------0

1234   1243   2134   2143

12 swap, 34 swap independently. Isomorphic with the rotation group of a cuboid, two pairs of opposite faces being labelled 12, 34. Subgroups and Normal subgroups: 2.01 4.07; Parent groups: 4.03 5.06 6.24 6.34 7.37; Normal supergroups: 4.03(8m) 6.24(8m) 6.34(8m) 7.20(48m) 7.22(24m) 7.25(24m) 7.34(24m) 7.37(12m).


0----------0
| 2,2  N=1 |     [4.07]   Order 2p
0----------0

1234   2143

12 swap, 34 swap to keep +ve parity. Subgroups: none; Parent groups: 4.04 4.05 4.06 5.04 5.07 6.26 6.27 6.32 6.35 6.36 7.39; Normal supergroups: 4.03(8m) 4.04(4p) 4.05(4m) 4.06(4m) 6.20(16m) 6.21(8m) 6.22(8m) 6.23(8p) 6.24(8m) 6.25(8m) 6.26(4p) 6.27(4m) 6.34(8m) 6.35(4p) 6.36(4m) 7.20(48m) 7.22(24m) 7.23(24p) 7.24(24m) 7.25(24m) 7.26(24m) 7.27(24m) 7.29(12m) 7.30(12p) 7.31(12p) 7.32(12m) 7.34(24m) 7.35(12p) 7.36(12m) 7.37(12m) 7.39(6p).


5 working bells, transitive

0---------------0
| 5        N=24 |     [5.01]   Order 120m
| 4,1      N=30 |
| 3,2      N=20 |
| 3,1,1    N=20 |
| 2,2,1    N=15 |
| 2,1,1,1  N=10 |
0---------------0

The extent on 5 bells. Primary subgroups: 4.01 5.02 5.03 5.06; Normal subgroup: 5.02; Parent groups: 6.01 7.08; Normal supergroup: 7.08(240m).


0---------------0
| 5        N=24 |     [5.02]   Order 60p
| 3,1,1    N=20 |
| 2,2,1    N=15 |
0---------------0

The Alternating group; all the +ve permutations of 12345. Isomorphic with the icosahedral rotation group, sets of mutually parallel or perpendicular edges labelled 1 to 5. Primary subgroups: 4.02 5.04 5.07; Normal subgroups: none; Parent groups: 5.01 6.02 7.09 7.10; Normal supergroups: 5.01(120m) 7.08(240m) 7.09(120p) 7.10(120m).


0-------------0
| 5      N=4  |     [5.03]   Order 20m
| 4,1    N=10 |
| 2,2,1  N=5  |
0-------------0

12345 23451 34512 45123 51234
54321 43215 32154 21543 15432
13524 35241 52413 24135 41352
42531 25314 53142 31425 14253

Cycle of 12345, with three involutions. Isomorphic with the rotation-translation group of an infinite square lattice, vertices labelled regularly 1 to 5 (as knight's move in chess). Primary subgroups: 4.05 5.04; Normal subgroups: 5.04 5.05; Parent groups: 5.01 6.03 7.11; Normal supergroup: 7.11(40m).


0------------0
| 5      N=4 |     [5.04]   Order 10p
| 2,2,1  N=5 |
0------------0

12345   23451   34512   45123   51234
15432   21543   32154   43215   54321

The dihedral group on 5 bells. A 5-part cycle and its reverse. Isomorphic with the rotational group of a pentagonal prism. Subgroups: 4.07 5.05; Normal subgroup: 5.05; Parent groups: 5.02 5.03 6.05 7.12 7.13; Normal supergroups: 5.03(20m) 7.11(40m) 7.12(20p) 7.13(20m).


0--------0
| 5  N=4 |     [5.05]   Order 5p
0--------0

12345   23451   34512   45123   51234

The cyclic group on 5 bells. Isomorphic with the rotation group of a regular pentagon. Subgroups: none; Parent groups: 5.04 7.14 7.15; Normal supergroups: 5.03(20m) 5.04(10p) 7.11(40m) 7.12(20p) 7.13(20m) 7.14(10m) 7.15(10m).


5 working bells, partitioned (3, 2)

0--------------0
| 3,2      N=2 |     [5.06]   Order 12m
| 3,1,1    N=2 |
| 2,2,1    N=3 |
| 2,1,1,1  N=4 |
0--------------0

12345   21345   23145   32145   31245   13245
12354   21354   23154   32154   31254   13254

123 permute, 45 swap independently. The group used in J.J.Parker's 12-part peal of Grandsire Triples. Primary subgroups: 3.01 4.06 5.07 5.08; Normal subgroups: 2.01* 3.01 3.02 5.07 5.08; Parent groups: 5.01 6.17 6.28 7.34; Normal supergroup: 7.34(24m).

* 2.01 is normal only if it swaps the isolated pair.


0------------0
| 3,1,1  N=2 |     [5.07]   Order 6p
| 2,2,1  N=3 |
0------------0

12345   21354   23145   32154   31245   13254

123 permute, 45 swap to keep parity. The rotation group of a triangular prism, the rectangular faces being labelled 1 to 3 and the end faces 4 and 5. The rows of a Stedman Doubles six (compare with group 7.40). Subgroups: 3.02 4.07; Normal subgroup: 3.02; Parent groups: 5.02 5.06 6.19 6.30 7.35 7.38; Normal supergroups: 5.06(12m) 7.34(24m) 7.35(12p) 7.38(12m).


0--------------0
| 3,2      N=2 |     [5.08]   Order 6m
| 3,1,1    N=2 |
| 2,1,1,1  N=1 |
0--------------0

12345   23145   31245   12354   23154   31254

123 rotate, 45 swap independently. Subgroups: 2.01 3.02; Parent groups: 5.06 6.18 6.29 7.37 7.38; Normal subgroups: 2.01 3.02Normal supergroups: 5.06(12m) 7.34(24m) 7.37(12m) 7.38(12m).


6 working bells, transitive

0------------------0
| 6          N=120 |     [6.01]   Order 720m
| 5,1        N=144 |
| 4,2        N=90  |
| 4,1,1      N=90  |
| 3,3        N=40  |
| 3,2,1      N=120 |
| 3,1,1,1    N=40  |
| 2,2,2      N=15  |
| 2,2,1,1    N=45  |
| 2,1,1,1,1  N=15  |
0------------------0

The extent on 6 bells. Primary subgroups: 5.01 6.02 6.03 6.04 6.06 6.17; Normal subgroup: 6.02; Parent group: 7.01; Normal supergroups: none.


0----------------0
| 5,1      N=144 |     [6.02]   Order360p
| 4,2      N=90  |
| 3,3      N=40  |
| 3,1,1,1  N=40  |
| 2,2,1,1  N=45  |
0----------------0

The Alternating Group on 6 bells. All the +ve permutations. Primary subgroups: 5.02 6.05 6.07 6.09 6.19; Normal subgroups: none; Parent groups: 6.01 7.02; Normal supergroup: 6.01(720m).


0---------------0
| 6        N=20 |     [6.03]   Order 120m
| 5,1      N=24 |
| 4,1,1    N=30 |
| 3,3      N=20 |
| 2,2,2    N=10 |
| 2,2,1,1  N=15 |
0---------------0

123456 124365 125643 126534 132546 135264 136425 142635 145326 154236
234561 243651 256431 265341 325461 352641 364251 426351 453261 542361
345612 436512 564312 653412 254613 526413 642513 263514 532614 423615
456123 365124 643125 534126 546132 264135 425136 635142 326145 236154
561234 651243 431256 341265 461325 641352 251364 351426 261453 361542
612345 512436 312564 412653 613254 413526 513642 514263 614532 615423

165432 156342 134652 143562 164523 146253 152463 153624 162354 163245
654321 563421 346521 435621 645231 462531 524631 536241 623541 632451
543216 634215 465213 356214 452316 625314 246315 362415 235416 324516
432165 342156 652134 562143 523164 253146 463152 624153 354162 245163
321654 421563 521346 621435 231645 531462 631524 241536 541623 451632
216543 215634 213465 214356 316452 314625 315246 415362 416235 516324

This group consists of one-sixth of the extent on 6 bells, and is mixed. Its positive rows give Hudson's group [6.05]. A commentary on this group is given after the group listings. Primary subgroups: 5.03 6.05 6.10 6.13; Normal subgroup: 6.05; Parent group: 6.01; Normal supergroups: none.


0-----------------0
| 6          N=12 |     [6.04]   Order 72m
| 4,2        N=18 |
| 3,3        N=4  |
| 3,2,1      N=12 |
| 3,1,1,1    N=4  |
| 2,2,2      N=6  |
| 2,2,1,1    N=9  |
| 2,1,1,1,1  N=6  |
0-----------------0

123456 163254 143652 123654 163452 143256
234561 632541 436521 236541 634521 432561
345612 325416 365214 365412 345216 325614
456123 254163 652143 654123 452163 256143
561234 541632 521436 541236 521634 561432
612345 416325 214365 412365 216345 614325

654321 452361 256341 456321 254361 652341
543216 523614 563412 563214 543612 523416
432165 236145 634125 632145 436125 234165
321654 361452 341256 321456 361254 341652
216543 614523 412563 214563 612543 416523
165432 145236 125634 145632 125436 165234

Cycle of 6 bells with 11 other cycles, made by rotating and inverting the two sets of 3 alternate bells. Primary subgroups: 6.07 6.08 6.24 6.28; Normal subgroups: 6.07 6.08 6.28 6.30 6.31; Parent group: 6.01; Normal supergroups: none.


0---------------0
| 5,1      N=24 |     [6.05]   Order 60p
| 3,3      N=20 |
| 2,2,1,1  N=15 |
0---------------0

123456 213546 312564 413625 514236 612435
134562 235461 325641 436251 542361 624351
145623 254613 356412 462513 523614 643512
156234 246135 364125 425136 536142 635124
162345 261354 341256 451362 561423 651243

154326 264531 346521 452631 563241 653421
143265 245316 365214 426315 532416 634215
132654 253164 352146 463152 524163 642153
126543 231645 321465 431526 541632 621534
165432 216453 314652 415263 516324 615342

Hudson's courses used for twin-bob composition in Stedman Triples. Isomorphic with the rotation group of the icosahedron, opposite pairs of vertices being labelled 1 to 6. Primary subgroups: 5.04 6.14 6.32; Normal subgroups: none; Parent groups: 6.02 6.03; Normal supergroup: 6.03(120m).


0----------------0
| 6          N=8 |     [6.06]   Order 48m
| 4,2        N=6 |
| 4,1,1      N=6 |
| 3,3        N=8 |
| 2,2,2      N=7 |
| 2,2,1,1    N=9 |
| 2,1,1,1,1  N=3 |
0----------------0

123456  341256  345612  563412  561234  125634
213456  431256  435612  653412  651234  215634
124356  342156  346512  564312  562134  126534
214356  432156  436512  654312  652134  216534
123465  341265  345621  563421  561243  125643
213465  431265  435621  653421  651243  215643
124365  342165  346521  564321  562143  126543
214365  432165  436521  654321  652143  216543

Pairs 12, 34, 56 permute, also swap independently. Primary subgroups: 6.09 6.10 6.11 6.13 6.20; Normal subgroups: 6.09 6.10 6.11 6.14 6.34 6.35 6.37*; Parent group: 6.01; Normal supergroups: none.

* Of the 7 elements of transformation kind (2,2,2) one has the three segregated pairs swapping: 214365. This, with rounds, forms the group 6.37 which is a normal subgroup of 6.06; for the other six elements the group formed is not a normal subgroup.


0---------------0
| 4,2      N=18 |     [6.07]   Order 36p
| 3,3      N=4  |
| 3,1,1,1  N=4  |
| 2,2,1,1  N=9  |
0---------------0

123456 163254 143652 236541 634521 432561
345612 325416 365214 654123 452163 256143
561234 541632 521436 412365 216345 614325
543216 523614 563412 456321 254361 652341
321654 361452 341256 632145 436125 234165
165432 145236 125634 214563 612543 416523

The +ve rows from group [6.04]. Primary subgroups: 6.26 6.30; Normal subgroups: 6.30 6.31; Parent groups: 6.02 6.04; Normal supergroup: 6.04(72m).


0---------------0
| 6        N=12 |     [6.08]   Order 36m
| 3,3      N=4  |
| 3,1,1,1  N=4  |
| 2,2,2    N=6  |
| 2,2,1,1  N=9  |
0---------------0

123456  163254  143652  654321  452361  256341
234561  632541  436521  165432  145236  125634
345612  325416  365214  216543  614523  412563
456123  254163  652143  321654  361452  341256
561234  541632  521436  432165  236145  634125
612345  416325  214365  543216  523614  563412

In a cycle of 6 bells, alternate bells are rotated, to give group [6.12]. All cycles also reversed. Primary subgroups: 6.12 6.13 6.30; Normal subgroups: 6.12 6.16 6.30 6.31 6.33; Parent group: 6.04; Normal supergroup: 6.04(72m).


0--------------0
| 4,2      N=6 |     [6.09]   Order 24p
| 3,3      N=8 |
| 2,2,1,1  N=9 |
0--------------0

123456  341256  345612  563412  561234  125634
214356  432156  436512  654312  652134  216534
213465  431265  435621  653421  651243  215643
124365  342165  346521  564321  562143  126543

Pairs 12, 34, 56 permute, each pair swapping independently but preserving parity. Isomorphic with the rotation group of a cube, opposite pairs of edges labelled 1 to 6. Used for the composition of Scientific Triples. Primary subgroups: 6.14 6.23 6.32; Normal subgroups: 6.14 6.35; Parent groups: 6.02 6.06 7.03; Normal supergroup: 6.06(48m).


0--------------0
| 4,1,1    N=6 |     [6.10]   Order 24m
| 3,3      N=8 |
| 2,2,2    N=6 |
| 2,2,1,1  N=3 |
0--------------0

123456  345612  561234  125643  341265  563421
214356  436512  652134  216543  432165  654321
213465  435621  651243  215634  431256  653412
124365  346521  562143  126534  342156  564312

Pairs 12, 34, 56 permute. Individual pairs swap, keeping parity in such a way that

(a) if perm of (12), (34), (56) +ve, then row +ve
(b) if perm of (12), (34), (56) -ve, then row -ve

Isomorphic with the rotation group of a cube, faces labelled 1 to 6. Primary subgroups: 6.14 6.16 6.25; Normal subgroups: 6.14 6.35; Parent groups: 6.03 6.06; Normal supergroup: 6.06(48m).


0------------------0
| 6          N = 8 |     [6.11]   Order 24m
| 3,3        N = 8 |
| 2,2,2      N = 1 |
| 2,2,1,1    N = 3 |
| 2,1,1,1,1  N = 3 |
0------------------0

123456   345612   561234
213456   435612   651234
124356   346512   562134
214356   436512   652134
123465   345621   561243
213465   435621   651243
124365   346521   562143
214365   436521   652143

Pairs 12, 34, 56 rotate, also swap independently. Primary subgroups: 6.14 6.15 6.34; Normal subgroups: 6.14 6.34 6.35 6.37; Parent group: 6.06; Normal supergroup: 6.06(48m).


0--------------0
| 6        N=6 |     [6.12]   Order 18m
| 3,3      N=4 |
| 3,1,1,1  N=4 |
| 2,2,2    N=3 |
0--------------0

123456  163254  143652
234561  632541  436521
345612  325416  365214
456123  254163  652143
561234  541632  521436
612345  416325  214365

The cyclic group of 6 bells, together with two other cycles, made by rotating a set of 3 alternate bells. Isomorphic with the rotation-translation group of an infinite lattice of regular hexagons, the vertices being labelled 1 to 6 in a regular fashion. Primary subgroups: 6.15 6.16 6.31; Normal subgroups: 6.16 6.31 6.33; Parent group: 6.08; Normal supergroup: 6.08(36m).


0--------------0
| 6        N=2 |     [6.13]   Order 12m
| 3,3      N=2 |
| 2,2,2    N=4 |
| 2,2,1,1  N=3 |
0--------------0

123456  234561  345612  456123  561234  612345
165432  216543  321654  432165  543216  654321

The dihedral group on 6 bells, formed of a cycle of six bells together with the reverse cycle. Isomorphic with the rotation group of a hexagonal prism, the rectangular faces (or six parallel edges) being labelled 1 to 6. Primary subgroups: 6.15 6.16 6.27 6.32; Normal subgroups: 6.15 6.16 6.32 6.33 6.37*; Parent groups: 6.03 6.06 6.08; Normal supergroups: none.

* Of the 4 elements of kind (2,2,2) in this group, only one has the three pairs of bells swapping which are mutually opposite in the hexagon form; only this one, with rounds, gives a normal subgroup.


0--------------0
| 3,3      N=8 |     [6.14]   Order 12p
| 2,2,1,1  N=3 |
0--------------0

123456  345612  561234
124365  346521  562143
213465  435621  651243
214356  436512  652134

Pairs 12, 34, 56 rotate and swap to keep parity. Isomorphic with the rotation group of a tetrahedron, opposite pairs of edges being labelled 1,2; 3,4; 5,6. Primary subgroup: 6.35; Subgroup: 6.33; Normal subgroup: 6.35; Parent groups: 6.05 6.09 6.10 6.11; Normal supergroups: 6.06(48m) 6.09(24p) 6.10(24m) 6.11(24m).


0------------0
| 6      N=2 |     [6.15]   Order 6m
| 3,3    N=2 |
| 2,2,2  N=1 |
0------------0

123456 234561 345612 456123 561234 612345        

The cyclic group on 6 bells. Isomorphic with the rotation group of a regular hexagon. Subgroups: 6.33 6.37; Normal subgroups: 6.33 6.37; Parent groups: 6.11 6.12 6.13 7.04; Normal supergroup: 6.13(12m).


0-----------0
| 3,3   N=2 |     [6.16]   Order 6m
| 2,2,2 N=3 |
0-----------0

123456   231564   312645    654321   546213   465132

Isomorphic with the rotation group of a triangular prism, the vertices being labelled 1 to 6. Subgroups: 6.33 6.37; Normal subgroup: 6.33; Parent groups: 6.10 6.12 6.13; Normal supergroups: 6.08(36m) 6.12(18m) 6.13(12m).


6 working bells, partitioned (4,2)

0----------------0
| 4,2        N=6 |     [6.17]   Order 48m
| 4,1,1      N=6 |
| 3,2,1      N=8 |
| 3,1,1,1    N=8 |
| 2,2,2      N=3 |
| 2,2,1,1    N=9 |
| 2,1,1,1,1  N=7 |
0----------------0

1234 permute, 56 swap independently. Primary subgroups: 4.01 5.06 6.18 6.19 6.20; Normal subgroups: 2.01* 4.01 4.02 4.04 6.18 6.19 6.21; Parent groups: 6.01 7.08 7.16; Normal supergroups: none.

* 2.01 is a normal subgroup only if the single pair of bells swapping is the independent pair.


0----------------0
| 3,2,1      N=8 |     [6.18]   Order 24m
| 3,1,1,1    N=8 |
| 2,2,2      N=3 |
| 2,2,1,1    N=3 |
| 2,1,1,1,1  N=1 |
0----------------0

1234 form the alternating group [4.02], 56 swap independently. Primary subgroups: 4.02 5.08 6.21; Normal subgroups: 2.01 4.02 4.04 6.21; Parent groups: 6.17 7.10 7.18; Normal supergroup: 6.17(48m).


0--------------0
| 4,2      N=6 |     [6.19]   Order 24p
| 3,1,1,1  N=8 |
| 2,2,1,1  N=9 |
0--------------0

1234 permute, 56 swap to keep +ve parity. Isomorphic with the rotation group of a cube. Primary subgroups: 4.02 5.07 6.23; Normal subgroups: 4.02 4.04; Parent groups: 6.02 6.17 7.09 7.19; Normal supergroup: 6.17(48m).


0----------------0
| 4,2        N=2 |     [6.20]   Order 16m
| 4,1,1      N=2 |
| 2,2,2      N=3 |
| 2,2,1,1    N=5 |
| 2,1,1,1,1  N=3 |
0----------------0

1234 form dihedral group [4.03], 56 swap independently. Primary subgroups: 4.03 6.21 6.22 6.23 6.24 6.25 6.34; Normal subgroups: 2.01* 4.03 4.04 4.05 4.07* 6.21 6.22 6.23 6.24 6.25 6.26 6.27* 6.34 6.35 6.36* 6.37*; Parent groups: 6.06 6.17 7.20; Normal supergroups: none.

* 2.01 is normal only if the swapping pair is independent. 4.07 is normal if the swapping pairs are "opposites" (see [4.03]). 6.27 is normal if the four bells "pairs of pairs" are from the dihedral group, and moreover particular pairs (see [6.27]) are "opposites". 6.36 needs the pairs swapping in phase to be "opposites". 6.37 needs two pairs of the three swapping as "opposites".


0----------------0
| 2,2,2      N=3 |     [6.21]   Order 8m
| 2,2,1,1    N=3 |
| 2,1,1,1,1  N=1 |
0----------------0

123456  214356  341256  432156
123465  214365  341265  432165

1234 form the `pairs of pairs' group [4.04], 56 swap independently. Primary subgroups: 4.04 6.27 6.36; Normal subgroups: 2.01 4.04 4.07 6.27 6.36 6.37; Parent groups: 6.18 6.20 7.27; Normal supergroups: 6.17(48m) 6.18(24m) 6.20(16m).


0----------------0
| 4,2        N=2 |     [6.22]   Order 8m
| 4,1,1      N=2 |
| 2,2,2      N=1 |
| 2,2,1,1    N=1 |
| 2,1,1,1,1  N=1 |
0----------------0

123456  234156  341256  412356
123465  234165  341265  412365

1234 rotate cyclically, 56 swap independently. Primary subgroups: 4.05 6.26 6.36; Normal subgroups: 2.01 4.05 4.07 6.26 6.36 6.37; Parent groups: 6.20 7.11 7.26; Normal supergroup: 6.20(16m).


0--------------0
| 4,2      N=2 |     [6.23]   Order 8p
| 2,2,1,1  N=5 |
0--------------0

123456   234165   341256   412365
143265   214356   321465   432156

1234 form the dihedral group [4.03], 56 swap to keep +ve parity. The rotation group of a square prism, side faces labelled 1 to 4 and opposite pairs of (shorter) parallel edges 5 and 6. Primary subgroups: 4.04 6.26 6.35; Normal subgroups: 4.04 4.07* 6.26 6.35; Parent groups: 6.09 6.19 6.20 7.23 7.28; Normal supergroup: 6.20(16m).

* If pairs are "opposites".


0----------------0
| 4,2        N=2 |     [6.24]   Order 8m
| 2,2,2      N=2 |
| 2,2,1,1    N=1 |
| 2,1,1,1,1  N=2 |
0----------------0

123456   234165   341256   412365
143256   214365   321456   432165

1234 form the dihedral group [4.03], 56 swap in linked form. The swapping above is linked to the positions of the pairs 13, 24, which remain segregated in [4.03] in pairs of positions. The rotation group of a square prism, side faces being labelled 1-4, opposite faces 1 and 3 labelled also 5, faces 2 and 4 labelled also 6. Primary subgroups: 4.06 6.26 6.27; Normal subgroups: 4.06 4.07 6.26 6.27; Parent groups: 6.04 6.20 7.25; Normal supergroup: 6.20(16m).


0----------------0
| 4,1,1      N=2 |     [6.25]   Order 8m
| 2,2,2      N=2 |
| 2,2,1,1    N=3 |
0----------------0

123456   234156   341256   412356
143265   214365   321465   432165

1234 form the dihedral group [4.03], 56 swap for the reverse cycle. The rotation group of a square prism, the side faces being labelled 1 to 4 and the square faces 5 and 6. Primary subgroups: 4.05 6.27 6.35; Normal subgroups: 4.05 4.07* 6.27 6.35; Parent groups: 6.10 6.20 7.24; Normal supergroup: 6.20(16m).

* 4.07 is normal when its pairs of swapping bells are "opposites"


0--------------0
| 4,2      N=2 |     [6.26]   Order 4p
| 2,2,1,1  N=1 |
0--------------0

123456   234165   341256   412365

1234 rotate cyclically, 56 swap to keep +ve parity. Subgroup: 4.07; Normal subgroup: 4.07; Parent groups: 6.07 6.22 6.23 6.24 7.12 7.30; Normal supergroups: 6.20(16m) 6.22(8m) 6.23(8p) 6.24(8m).


0--------------0
| 2,2,2    N=2 |     [6.27]   Order 4m
| 2,2,1,1  N=1 |
0--------------0

123456   214356   341265   432165

1234 form the "pairs of pairs" group [4.04], 56 swap in sympathy with the disposal of a particular pair of pairs (in the case given, 12/34). The "pairs of pairs" group is isomorphic with the rotation group of a cuboid, four equal edges being labelled 1 to 4. Here also, a pair of opposite faces are labelled 5 and 6. Subgroups: 4.07 6.37; Normal subgroups: 4.07 6.37; Parent groups: 6.13 6.21 6.24 6.25 7.32; Normal supergroups: 6.20(16m) 6.21(8m) 6.24(8m) 6.25(8m).


6 working bells, partitioned (3, 3)

0-----------------0
| 3,3        N=4  |     [6.28]   Order 36m
| 3,2,1      N=12 |
| 3,1,1,1    N=4  |
| 2,2,1,1    N=9  |
| 2,1,1,1,1  N=6  |
0-----------------0

123456 123546 123564 123654 123645 123465
213456 213546 213564 213654 213645 213465
231456 231546 231564 231654 231645 231465
321456 321546 321564 321654 321645 321465
312456 312546 312564 312654 312645 312465
132456 132546 132564 132654 132645 132465

123 permute, 456 permute independently. Primary subgroups: 5.06 6.29 6.30; Normal subgroups: 3.01 3.02 6.29 6.30 6.31; Parent groups: 6.04 7.16; Normal supergroup: 6.04(72m).


0----------------0
| 3,3        N=4 |     [6.29]   Order 18m
| 3,2,1      N=6 |
| 3,1,1,1    N=4 |
| 2,1,1,1,1  N=3 |
0----------------0

123456  213456  231456  321456  312456  132456
123564  213564  231564  321564  312564  132564
123645  213645  231645  321645  312645  132645

123 permute, 456 rotate independently. Primary subgroups: 3.01 5.08 6.31; Normal subgroups: 3.01 3.02 6.31; Parent groups: 6.28 7.17 7.18; Normal supergroup: 6.28(36m).


0--------------0
| 3,3      N=4 |     [6.30]   Order 18p
| 3,1,1,1  N=4 |
| 2,2,1,1  N=9 |
0--------------0

123456  213546  231456  321546  312456  132546
123564  213654  231564  321654  312564  132654
123645  213465  231645  321465  312645  132465

123 permute, 456 permute to keep +ve parity. Primary subgroups: 5.07 6.31 6.32; Normal subgroups: 3.02 6.31 6.33; Parent groups: 6.07 6.08 6.28 7.19; Normal supergroups: 6.04(72m) 6.07(36p) 6.08(36m) 6.28(36m).


0--------------0
| 3,3      N=4 |     [6.31]   Order 9p
| 3,1,1,1  N=4 |
0--------------0

123456  231456  312456
123564  231564  312564
123645  231645  312645

123 rotate cyclically, 564 rotate independently. Subgroups: 3.02 6.33; Normal subgroups: 3.02 6.33; Parent groups: 6.12 6.29 6.30 7.21; Normal supergroups: 6.04(72m) 6.07(36p) 6.08(36m) 6.12(18m) 6.28(36m) 6.29(18m) 6.30(18p).


0--------------0
| 3,3      N=2 |     [6.32]   Order 6p
| 2,2,1,1  N=3 |
0--------------0

123456  213546  231564   321654  312645  132465

123 permute, 456 permute in phase. Subgroups: 4.07 6.33; Normal subgroup: 6.33; Parent groups: 6.05 6.09 6.13 6.30 7.28; Normal supergroup: 6.13(12m).


0----------0
| 3,3  N=2 |     [6.33]   Order 3p
0----------0

123456  231564  312645                             

123 rotate cyclically, 456 rotate in phase. Subgroups: none; Parent groups: 6.14 6.15 6.16 6.31 6.32 7.05 7.33; Normal supergroups: 6.08(36m) 6.12(18m) 6.13(12m) 6.15(6m) 6.16(6m) 6.30(18p) 6.31(9p) 6.32(6p) .


6 working bells, partitioned (2, 2, 2)

0----------------0
| 2,2,2      N=1 |     [6.34]   Order 8m
| 2,2,1,1    N=3 |
| 2,1,1,1,1  N=3 |
0----------------0

123456  213456  124356  214356
123465  213465  124365  214365

Pairs 12, 34, 56 swap independently. Primary subgroups: 4.06 6.35 6.36; Normal subgroups: 2.01 4.06 4.07 6.35 6.36 6.37; Parent groups: 6.11 6.20 7.34; Normal supergroups: 6.06(48m) 6.11(24m) 6.20(16m).


0---------------0
| 2,2,1,1   N=3 |     [6.35]   Order 4p
0---------------0

123456  214356  124365  213465      

Pairs 12, 34, 56 swap, preserving +ve parity. Isomorphic with the rotation group of a cuboid, opposite pairs of faces being labelled 1 & 2, 3 & 4, 5 & 6. Subgroup: 4.07; Normal subgroup: 4.07; Parent groups: 6.14 6.23 6.25 6.34 7.35; Normal supergroups: 6.06(48m) 6.09(24p) 6.10(24m) 6.11(24m) 6.14(12p) 6.20(16m) 6.23(8p) 6.25(8m) 6.34(8m).


0----------------0
| 2,2,2      N=1 |     [6.36]   Order 4m
| 2,2,1,1    N=1 |
| 2,1,1,1,1  N=1 |
0----------------0

123456  214356  123465  214365

Pairs 12, 34 swap in phase, 56 swap independently. Subgroups: 2.01 4.07 6.37; Normal subgroups: 2.01 4.07 6.37; Parent groups: 6.21 6.22 6.34 7.13 7.36 7.38; Normal supergroups: 6.20(16m) 6.21(8m) 6.22(8m) 6.34(8m).


0------------0
| 2,2,2  N=1 |     [6.37]   Order 2m
0------------0

123456  214365

Pairs 12, 34, 56 swap in phase. Subgroups: none; Parent groups: 6.15 6.16 6.27 6.36 7.06 7.15 7.40; Normal supergroups: 6.06(48m) 6.11(24m) 6.13(12m) 6.15(6m) 6.20(16m) 6.21(8m) 6.22(8m) 6.27(4m) 6.34(8m) 6.36(4m).


7 working bells, transitive

0--------------------0
| 7            N=720 |     [7.01]   Order 5040m
| 6,1          N=840 |
| 5,2          N=504 |
| 5,1,1        N=504 |
| 4,3          N=420 |
| 4,2,1        N=630 |
| 4,1,1,1      N=210 |
| 3,3,1        N=280 |
| 3,2,2        N=210 |
| 3,2,1,1      N=420 |
| 3,1,1,1,1    N=70  |
| 2,2,2,1      N=105 |
| 2,2,1,1,1    N=105 |
| 2,1,1,1,1,1  N=21  |
0--------------------0

The extent on 7 bells. Primary subgroups: 6.01 7.02 7.04 7.08 7.16; Normal subgroup: 7.02; Parent groups: none; Normal supergroups: none.


0--------------------0
| 7            N=720 |     [7.02]   Order 2520p
| 5,1,1        N=504 |
| 4,2,1        N=630 |
| 3,3,1        N=280 |
| 3,2,2        N=210 |
| 3,1,1,1,1    N=70  |
| 2,2,1,1,1    N=105 |
0--------------------0

All the +ve permutations. The Alternating group. Primary subgroups: 6.02 7.03 7.09 7.19; Normal subgroups: none; Parent group: 7.01; Normal supergroup: 7.01(5040m).


0-----------------0
| 7          N=48 |     [7.03]   Order 168p
| 4,2,1      N=42 |
| 3,3,1      N=56 |
| 2,2,1,1,1  N=21 |
0-----------------0

1234567   1325467   1452367   1543267   1674523   1765423
1235476   1324576   1453276   1542376   1675432   1764532
1236745   1326754   1456723   1546732   1672345   1762354
1237654   1327645   1457632   1547623   1673254   1763245

2136547   2315647   2473156   2567413   2651347   2745613
2135674   2316574   2471365   2564731   2653174   2746531
2134765   2314756   2475631   2561374   2654713   2741356
2137456   2317465   2476513   2563147   2657431   2743165

3126457   3214657   3462157   3572146   3641257   3751246
3124675   3216475   3461275   3571264   3642175   3752164
3125764   3215746   3465721   3574621   3645712   3754612
3127546   3217564   3467512   3576412   3647521   3756421

4156327   4271563   4365127   4513627   4631527   4725163
4153672   4275136   4361572   4516372   4635172   4721536
4152763   4276315   4362751   4512736   4632715   4726351
4157236   4273651   4367215   4517263   4637251   4723615

5146237   5264137   5374126   5412637   5621437   5731426
5142673   5261473   5371462   5416273   5624173   5734162
5143762   5263741   5372641   5413726   5623714   5732614
5147326   5267314   5376214   5417362   5627341   5736241

6172543   6254317   6345217   6432517   6523417   6715243
6175234   6253471   6342571   6435271   6524371   6712534
6174325   6251743   6341752   6431725   6521734   6714352
6173452   6257134   6347125   6437152   6527143   6713425

7162453   7241653   7354216   7426153   7532416   7614253
7164235   7246135   7352461   7421635   7534261   7612435
7165324   7245316   7351642   7425361   7531624   7615342
7163542   7243561   7356124   7423516   7536142   7613524

The group on which Scientific Triples is based. Primary subgroups: 6.09 7.05 7.28; Normal subgroups: none; Parent group: 7.02; Normal supergroups: none.


O---------------O
| 7        N=6  |     [7.04]   Order 42m
| 6,1      N=14 |
| 3,3,1    N=14 |
| 2,2,2,1  N=7  |
O---------------O

1234567 1357246 1526374 7654321 6427531 4736251
2345671 3572461 5263741 6543217 4275316 7362514
3456712 5724613 2637415 5432176 2753164 3625147
4567123 7246135 6374152 4321765 7531642 6251473
5671234 2461357 3741526 3217654 5316427 2514736
6712345 4613572 7415263 2176543 3164275 5147362
7123456 6135724 4152637 1765432 1642753 1473625

A cycle of 7 bells and all its involutes. Isomorphic with the rotation-translation group of a plane triangular grid, vertices labelled 1-7 regularly (each figure distributed in a larger triangular grid), rotations multiples of 60 degrees. The perms may be placed in the corners of the triangles. Primary subgroups: 6.15 7.05 7.06; Normal subgroups: 7.05 7.06 7.07; Parent group: 7.01; Normal supergroups: none.


O-------------O
| 7      N=6  |     [7.05]   Order 21p
| 3,3,1  N=14 |
O-------------O

1234567    1357246    1526374
3456712    5724613    5263741
2345671    3572461    2637415
4567123    7246135    6374152
5671234    2461357    3741526
6712345    4613572    7415263
7123456    6135724    4152637

A cycle of 7 bells and the even involutes. Isomorphic with a triangular grid as [7.04], but rotations of 120 and 240 degrees only, perms in corners of alternate triangles only. Subgroups: 6.33 7.07; Normal subgroup: 7.07; Parent groups: 7.03 7.04; Normal supergroup: 7.04(42m),


O--------------O
| 7        N=6 |     [7.06]   Order 14m
| 2,2,2,1  N=7 |
O--------------O

1234567 2345671 3456712 4567123 5671234 6712345 7123456
7654321 6543217 5432176 4321765 3217654 2176543 1765432

The dihedral group on 7 bells; a cycle and its reverse. Isomorphic with the rotation group of a heptagonal prism. Subgroups: 6.37 7.07; Normal subgroup: 7.07; Parent group: 7.04; Normal supergroup: 7.04(42m).


O--------O
| 7  N=6 |     [7.07]   Order 7p
O--------O

1234567  2345671  3456712  4567123  5671234  6712345  7123456

The cyclic group on 7 bells. Isomorphic with the rotation group of a regular heptagon. Subgroups: none; Parent groups: 7.05 7.06; Normal supergroups: 7.04(42m) 7.05(21p) 7.06(14m).


7 working bells, partitioned (5,2)

O-------------------O
| 5,2          N=24 |     [7.08]   Order 240m
| 5,1,1        N=24 |
| 4,2,1        N=30 |
| 4,1,1,1      N=30 |
| 3,2,2        N=20 |
| 3,2,1,1      N=40 |
| 3,1,1,1,1    N=20 |
| 2,2,2,1      N=15 |
| 2,2,1,1,1    N=25 |
| 2,1,1,1,1,1  N=11 |
O-------------------O

12345 permute forming [5.01], 67 swap independently. Primary subgroups: 5.01 6.17 7.09 7.10 7.11 7.34; Normal subgroups: 2.01* 5.01 5.02 7.09 7.10; Parent group: 7.01; Normal supergroups: none.

* 2.01 is a normal subgroup only when the swapping pair is the independent pair.


O-----------------O
| 5,1,1      N=24 |     [7.09]   Order 120p
| 4,2,1      N=30 |
| 3,2,2      N=20 |
| 3,1,1,1,1  N=20 |
| 2,2,1,1    N=25 |
O-----------------O

12345 permute forming [5.01], 67 swap to keep +ve parity. Primary subgroups: 5.02 6.19 7.12 7.35; Normal subgroup: 5.02; Parent groups: 7.02 7.08; Normal supergroup: 7.08(240m).


O-------------------O
| 5,2          N=24 |     [7.10]   Order 120m
| 5,1,1        N=24 |
| 3,2,1,1      N=20 |
| 3,1,1,1,1    N=20 |
| 2,2,2,1      N=15 |
| 2,2,1,1,1    N=15 |
| 2,1,1,1,1,1  N=1  |
O-------------------O

The alternating group [5.02] on 12345, 67 swap independently. Primary subgroups: 5.02 6.18 7.13 7.38; Normal subgroups: 2.01 5.02; Parent group: 7.08; Normal supergroup: 7.08(240m).


O-------------------O
| 5,2          N=4  |     [7.11]   Order 40m
| 5,1,1        N=4  |
| 4,2,1        N=10 |
| 4,1,1,1      N=10 |
| 2,2,2,1      N=5  |
| 2,2,1,1,1    N=5  |
| 2,1,1,1,1,1  N=1  |
O-------------------O

1234567   1352467   1425367   1543267
2345167   3524167   4253167   5432167
3451267   5241367   2531467   4321567
4512367   2413567   5314267   3215467
5123467   4235267   3142567   2154367
1234576   1352476   1425376   1543276
2345176   3524176   4253176   5432176
3451276   5241376   2531476   4321576
4512376   2413576   5314276   3215476
5123476   4235276   3142576   2154376

12345 perform a cycle and its 3 involutions forming group [5.03], 67 swap independently. Primary subgroups: 5.03 6.22 7.12 7.13; Normal subgroups: 2.01 5.03 5.04 5.05 7.12 7.13 7.14 7.15; Parent group: 7.08; Normal supergroups: none.


O-----------------O
| 5,1,1      N=4  |     [7.12]   Order 20p
| 4,2,1      N=10 |
| 2,2,1,1,1  N=5  |
O-----------------O

1234567  1352476  1425376  1543267
2345167  3524176  4253176  5432167
3451267  5241376  2531476  4321567
4512367  2413576  5314276  3215467
5123467  4235276  3142576  2154367

12345 perform a cycle and its 3 involutions forming group [5.03], 67 swap to keep +ve parity. Primary subgroups: 5.04 6.26; Normal subgroups: 5.04 5.05; Parent groups: 7.09 7.11; Normal supergroup: 7.11(40m).


O------------------O
| 5,2          N=4 |     [7.13]   Order 20m
| 5,1,1        N=4 |
| 2,2,2,1      N=5 |
| 2,2,1,1,1    N=5 |
| 2,1,1,1,1,1  N=1 |
O------------------O

1234567   1234576   1543276   1543267
2345167   2345176   5432176   5432167
3451267   3451276   4321576   4321567
4512367   4512376   3215476   3215467
5123467   5123476   2154376   2154367

The dihedral group [5.04] on 12345, 67 swapping independently. Primary subgroups: 5.04 6.36 7.14 7.15; Normal subgroups: 2.01 5.04 5.05 7.14 7.15; Parent groups: 7.10 7.11; Normal supergroup: 7.11(40m).


O------------------O
| 5,2          N=4 |     [7.14]   Order 10m
| 5,1,1        N=4 |
| 2,1,1,1,1,1  N=1 |
O------------------O

1234567  2345167  3451267  4512367  5123467
1234576  2345176  3451276  4512376  5123476

The cyclic group [5.05] on 12345, 67 swap independently. Subgroups: 2.01 5.05; Normal subgroups: 2.01 5.05; Parent group: 7.13; Normal supergroups: 7.11(40m) 7.13(20m).


O---------------O
| 5,1,1     N=4 |     [7.15]   Order 10m
| 2,2,2,1   N=5 |
O---------------O

1234567  2345167  3451267  4512367  5123467
1543276  2154376  3215476  4321576  5432176

The dihedral group [5.04] on 12345, 67 swap for the reverse cycle. The rotational group of a pentagonal prism, side faces labelled 1 to 5 and end faces labelled 6, 7. Subgroups: 5.05 6.37; Normal subgroup: 5.05; Parent group: 7.13; Normal supergroups: 7.11(40m) 7.13(20m).


7 working bells, partitioned (4,3)

O-------------------O
| 4,3          N=12 |     [7.16]   Order 144m
| 4,2,1        N=18 |
| 4,1,1,1      N=6  |
| 3,3,1        N=16 |
| 3,2,2        N=6  |
| 3,2,1,1      N=36 |
| 3,1,1,1,1    N=10 |
| 2,2,2,1      N=9  |
| 2,2,1,1,1    N=21 |
| 2,1,1,1,1,1  N=9  |
O-------------------O

The extent [4.01] on 1234, independent extent [3.01] on 567. Primary subgroups: 6.17 6.28 7.17 7.18 7.19 7.20; Normal subgroups: 3.01* 3.02* 4.01 4.02 4.04 7.17 7.18 7.19 7.21 7.27 7.31; Parent group: 7.01; Normal supergroups: none.

* Subgroups 3.01, 3.02 are normal only if they are concerned with the set of 3 bells.


O-------------------O
| 4,3          N=12 |     [7.17]   Order 72m
| 4,1,1,1      N=6  |
| 3,3,1        N=16 |
| 3,2,2        N=6  |
| 3,2,1,1      N=12 |
| 3,1,1,1,1    N=10 |
| 2,2,1,1,1    N=3  |
| 2,1,1,1,1,1  N=6  |
O-------------------O

The extent [4.01] on 1234, independent cyclic group [3.02] on 567. Primary subgroups: 4.01 6.29 7.21 7.22; Normal subgroups: 3.02* 4.01 4.02 4.04 7.21 7.31; Parent group: 7.16; Normal supergroup: 7.16(144m).

* Subgroup 3.02 is normal only when it concerns the set of 3 bells.


O-------------------O
| 3,3,1        N=16 |     [7.18]   Order 72m
| 3,2,2        N=6  |
| 3,2,1,1      N=24 |
| 3,1,1,1,1    N=10 |
| 2,2,2,1      N=9  |
| 2,2,1,1,1    N=3  |
| 2,1,1,1,1,1  N=3  |
O-------------------O

The alternating group [4.02] on 1234, independent extent [3.01] on 567. Primary subgroups: 6.18 6.29 7.21 7.27; Normal subgroups: 3.01 3.02* 4.02 4.04 7.21 7.27 7.31; Parent group: 7.16; Normal supergroup: 7.16(144m).

* Subgroup 3.02 is normal only when it concerns the set of 3 bells.


O-----------------O
| 4,2,1      N=18 |     [7.19]   Order 72p
| 3,3,1      N=16 |
| 3,2,2      N=6  |
| 3,1,1,1,1  N=10 |
| 2,2,1,1,1  N=21 |
O-----------------O

The extent [4.01] on 1234, with independent extent on 567 but keeping +ve parity. Primary subgroups: 6.19 6.30 7.21 7.23 7.28; Normal subgroups: 3.02* 4.02 4.04 7.21 7.31 7.33; Parent groups: 7.02 7.16; Normal supergroup: 7.16(144m).

* Normal only when it concerns the set of 3 bells.


O------------------O
| 4,3          N=4 |     [7.20]   Order 48m
| 4,2,1        N=6 |
| 4,1,1,1      N=2 |
| 3,2,2        N=6 |
| 3,2,1,1      N=4 |
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=9 |
| 2,2,1,1,1    N=9 |
| 2,1,1,1,1,1  N=5 |
O------------------O

The dihedral group [4.03] on 1234, independent extent [3.01] on 567. Primary subgroups: 6.20 7.22 7.23 7.24 7.25 7.26 7.27 7.34; Normal subgroups: 3.01 3.02 4.03 4.04 4.05 4.06* 4.07* 7.22 7.23 7.24; 7.25 7.26 7.27 7.29 7.30 7.31 7.32* 7.34 7.35 7.36* 7.37 7.39* 7.40*; Parent group: 7.16; Normal supergroups: none.

* In each case, normality depends on a pair of pairs being "opposites" (see [4.03]); for 7.32 linked pairs; for 7.39 the independent pairs.


O-----------------O
| 3,3,1      N=16 |     [7.21]   Order 36p
| 3,2,2      N=6  |
| 3,1,1,1,1  N=10 |
| 2,2,1,1,1  N=3  |
O-----------------O

The alternating group [4.02] on 1234, independent cyclic group [3.02] on 567. Primary subgroups: 4.02 6.31 7.31 7.33; Normal subgroups: 3.02* 4.02 4.04 7.31 7.33; Parent groups: 7.17 7.18 7.19; Normal supergroups: 7.16(144m) 7.17(72m) 7.18(72m) 7.19(72p).

* normal only when it concerns the set of 3 bells


O------------------O
| 4,3          N=4 |     [7.22]   Order 24m
| 4,1,1,1      N=2 |
| 3,2,2        N=6 |
| 3,2,1,1      N=4 |
| 3,1,1,1,1    N=2 |
| 2,2,1,1,1    N=3 |
| 2,1,1,1,1,1  N=2 |
O------------------O

1234567  1234675  1234756
2341567  2341675  2341756
3412567  3412675  3412756
4123567  4123675  4123756
4321567  4321675  4321756
3214567  3214675  3214756
2143567  2143675  2143756
1432567  1432675  1432756

The dihedral group [4.03] on 1234, independent cyclic group [3.02] on 567. Primary subgroups: 4.03 7.29 7.31 7.37; Normal subgroups: 3.02 4.03 4.04 4.05 4.06 4.07* 7.29 7.31 7.37 7.39*; Parent groups: 7.17 7.20; Normal supergroup: 7.20(48m).

* For conditions see [7.20].


O----------------O
| 4,2,1      N=6 |     [7.23]   Order 24p
| 3,2,2      N=6 |
| 3,1,1,1,1  N=2 |
| 2,2,1,1,1  N=9 |
O----------------O

1234567  1234675  1234756
2341657  2341765  2341576
3412567  3412675  3412756
4123657  4123765  4123576
4321567  4321675  4321756
3214657  3214765  3214576
2143567  2143675  2143756
1432657  1432765  1432576

The dihedral group [4.03] on 1234, with linked extent on 567 to preserve +ve parity. Primary subgroups: 6.23 7.30 7.31 7.35; Normal subgroups: 3.02 4.04 4.07* 7.30 7.31 7.35 7.39*; Parent groups: 7.19 7.20; Normal supergroup: 7.20(48m).

* For conditions see group [7.20].


O----------------O
| 4,3        N=4 |     [7.24]   Order 24m
| 4,1,1,1    N=2 |
| 3,2,2      N=2 |
| 3,1,1,1,1  N=2 |
| 2,2,2,1    N=6 |
| 2,2,1,1,1  N=7 |
O----------------O

1234567  1234675  1234756
2341567  2341675  2341756
3412567  3412675  3412756
4123567  4123675  4123756
4321657  4321765  4321576
3214657  3214765  3214576
2143657  2143765  2143576
1432657  1432765  1432576

The dihedral group [4.03] on 1234, with linked extent on 567. The +ve rows on 567 link with the "direct" cycle of 1234, the -ve rows with the reverse cycle. Primary subgroups: 6.25 7.29 7.32 7.35 7.35 7.39; Normal subgroups: 3.02 4.05 4.07* 7.29 7.32; Parent group: 7.20; Normal supergroup: 7.20(48m).

* For condition see group [7.20]


O------------------O
| 4,2,1        N=6 |     [7.25]   Order 24m
| 3,2,2        N=2 |
| 3,2,1,1      N=4 |
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=6 |
| 2,2,1,1,1    N=1 |
| 2,1,1,1,1,1  N=2 |
O------------------O

1234567  1234675  1234756
2341657  2341765  2341576
3412567  3412675  3412756
4123657  4123765  4123576
4321657  4321765  4321576
3214567  3214675  3214756
2143657  2143765  2143576
1432567  1432675  1432756

The dihedral group [4.03] on 1234, with linked extent on 567. In the above example, the +ve rows on 567 link with bells 1,3 in positions 1 and 3, the -ve rows with 1,3 in positions 2 and 4. Primary subgroups: 6.24 7.30 7.32 7.37; Normal subgroups: 3.02 4.06 4.07 7.30 7.32 7.37 7.39; Parent group: 7.20; Normal supergroup: 7.20(48m).


O------------------O
| 4,3          N=4 |     [7.26]   Order 24m
| 4,2,1        N=6 |
| 4,1,1,1      N=2 |
| 3,2,2        N=2 |
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=3 |
| 2,2,1,1,1    N=1 |
| 2,1,1,1,1,1  N=3 |
O------------------O

1234567  1234675  1234756
2341567  2341675  2341756
3412567  3412675  3412756
4123567  4123675  4123756

1234657  1234765  1234576
2341657  2341765  2341576
3412657  3412765  3412576
4123657  4123765  4123576

The cyclic group [4.05] on 1234, independent extent [3.01] on 567. Primary subgroups: 6.22 7.29 7.30 7.36; Normal subgroups: 3.01 3.02 4.05 4.07 7.29 7.30 7.36 7.39 7.40; Parent group: 7.20; Normal supergroup: 7.20(48m).


O------------------O
| 3,2,2        N=6 |     [7.27]   Order 24m
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=9 |
| 2,2,1,1,1    N=3 |
| 2,1,1,1,1,1  N=3 |
O------------------O

1234567  1234675  1234756
2143567  2143675  2143756
3412567  3412675  3412756
4321567  4321675  4321756

1234657  1234765  1234576
2143657  2143765  2143576
3412657  3412765  3412576
4321657  4321765  4321576

The pairs of pairs group [4.04] on 1234, independent extent [3.01] on 567. Primary subgroups: 6.21 7.31 7.32 7.36; Normal subgroups: 3.01 3.02 4.04 4.07 7.31 7.32 7.36 7.39 7.40; Parent groups: 7.18 7.20; Normal supergroups: 7.16(144m) 7.18(72m) 7.20(48m).


O----------------O
| 4,2,1      N=6 |     [7.28]   Order 24p
| 3,3,1      N=8 |
| 2,2,1,1,1  N=9 |
O----------------O

1234567  1423756  3241756
2341576  4231765  2413765
3412567  2314756  4132756
4123576  3142765  1324765
1342675  2431675  4321567
3421657  4312657  3214576
4213675  3124675  2143567
2134657  1243657  1432576

The extent [4.01] on 1234, with linked extent on 567. Isomorphic with the cubic rotation group, pairs of opposite vertices being labelled 1 to 4, and pairs of opposite faces labelled 5 to 7. Primary subgroups: 6.23 6.32 7.33; Normal subgroups: 4.04 7.33; Parent groups: 7.03 7.19; Normal supergroups: none.

There are two distinct subgroups given by 4.07; in one, both swapping bells are in the 4-set; in the other, one swapping pair is in the 3-set and the other in the 4-set; but neither is normal (cf. 4.03).


O----------------O
| 4,3        N=4 |     [7.29]   Order 12m
| 4,1,1,1    N=2 |
| 3,2,2      N=2 |
| 3,1,1,1,1  N=2 |
| 2,2,1,1,1  N=1 |
O----------------O

1234567  1234675  1234756
2341567  2341675  2341756
3412567  3412675  3412756
4123567  4123675  4123756

The cyclic group [4.05] on 1234, independent cyclic group [3.02] on 567. Primary subgroups: 4.05 7.39; Normal subgroups: 3.02 4.05 4.07 7.39; Parent groups: 7.22 7.24 7.26; Normal supergroups: 7.20(48m) 7.22(24m) 7.24(24m) 7.26(24m).


O----------------O
| 4,2,1      N=6 |     [7.30]   Order 12p
| 3,2,2      N=2 |
| 3,1,1,1,1  N=2 |
| 2,2,1,1,1  N=1 |
O----------------O

1234567  1234675  1234756
2341657  2341765  2341576
3412567  3412675  3412756
4123657  4123765  4123576

The cyclic group [4.05] on 1234, linked extent on 567 preserving +ve parity. Primary subgroups: 6.26 7.39; Normal subgroups: 3.02 4.07 7.39; Parent groups: 7.23 7.25 7.26; Normal supergroups: 7.20(48m) 7.23(24p) 7.25(24m) 7.26(24m).


O----------------O
| 3,2,2      N=6 |     [7.31]   Order 12p
| 3,1,1,1,1  N=2 |
| 2,2,1,1,1  N=3 |
O----------------O

1234567  1234675  1234756
2143567  2143675  2143756
3412567  3412675  3412756
4321567  4321675  4321756

The pairs of pairs group [4.04] on 1234, independent cyclic group [3.02] on 567. Primary subgroups: 4.04 7.39; Normal subgroups: 3.02 4.04 4.07 7.39; Parent groups: 7.21 7.22 7.23 7.27; Normal supergroups: 7.16(144m) 7.17(72m) 7.18(72m) 7.19(72p) 7.20(48m) 7.21(36p) 7.22(24m) 7.23(24p) 7.27(24m).


O----------------O
| 3,2,2      N=2 |     [7.32]   Order 12m
| 3,1,1,1,1  N=2 |
| 2,2,2,1    N=6 |
| 2,2,1,1,1  N=1 |
O----------------O

1234567  1234675  1234756
2143567  2143675  2143756
3412657  3412765  3412576
4321657  4321765  4321576

The pairs of pairs group [4.04] on 1234, linked extent on 567. The perms of 567 are linked to a particular pair of 1234 in a particular pair of places; here, +ve ones to 1 & 2 in 1sts & 2nds. Primary subgroups: 6.27 7.39 7.40; Normal subgroups: 3.02 4.07 7.39 7.40; Parent groups: 7.24 7.25 7.27; Normal supergroups: 7.20(48m) 7.24(24m) 7.25(24m) 7.27(24m).


O----------------O
| 3,3,1      N=8 |     [7.33]   Order 12p
| 2,2,1,1,1  N=3 |
O----------------O

1234567  1423756  3241756
3412567  2314756  4132756
1342675  2431675  4321567
4213675  3124675  2143567

The alternating group [4.02] on 1234, linked to the cyclic group [3.02] on 567. Isomorphic with the rotation group of the tetrahedron, with vertices labelled 1 to 4 and pairs of opposite edges 5 to 7. The signatures of [7.33] and [6.14] differ only in an extra working bell. Primary subgroup: 4.04; Subgroup: 6.33; Normal subgroup: 4.04; Parent groups: 7.21 7.28; Normal supergroups: 7.19(72p) 7.21(36p) 7.28(24p).


7 working bells, partitioned (3,2,2)

O------------------O
| 3,2,2        N=2 |     [7.34]   Order 24m
| 3,2,1,1      N=4 |
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=3 |
| 2,2,1,1,1    N=7 |
| 2,1,1,1,1,1  N=5 |
O------------------O

The extent [3.01] on 123, pairs 45 and 67 swap independently. Primary subgroups: 5.06 6.34 7.35 7.36 7.37 7.38; Normal subgroups: 2.01* 3.01 3.02 4.06* 4.07* 5.06 5.07 5.08 7.35 7.36 7.37 7.38 7.39 7.40; Parent groups: 7.08 7.20; Normal supergroup: 7.20(48m).

* for normality the three permuting bells must be fixed.


O----------------O
| 3,2,2      N=2 |     [7.35]   Order 12p
| 3,1,1,1,1  N=2 |
| 2,2,1,1,1  N=7 |
O----------------O

1234567  1235476  2134576  2135467
2314567  2315476  3214576  3215467
3124567  3125476  1324576  1325467

The extent [3.01] on 123, pairs 45 and 67 swap to keep +ve parity. Primary subgroups: 5.07 6.35 7.39; Normal subgroups: 3.02 4.07* 5.07 7.39; Parent groups: 7.09 7.23 7.24 7.34; Normal supergroups: 7.20(48m) 7.23(24p) 7.24(24m) 7.34(24m).

* for normality, the three permuting bells must be fixed.


O------------------O
| 3,2,2        N=2 |     [7.36]   Order 12m
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=3 |
| 2,2,1,1,1    N=1 |
| 2,1,1,1,1,1  N=3 |
O------------------O

1234567  1235476  2134567  2135476
2314567  2315476  3214567  3215476
3124567  3125476  1324567  1325476

The extent [3.01] on 123, pairs 45 and 67 swap in phase with each other. Primary subgroups: 3.01 6.36 7.39 7.40; Normal subgroups: 3.01 3.02 4.07 7.39 7.40; Parent groups: 7.26 7.27 7.34; Normal supergroups: 7.20(48m) 7.26(24m) 7.27(24m) 7.34(24m).


O------------------O
| 3,2,2        N=2 |     [7.37]   Order 12m
| 3,2,1,1      N=4 |
| 3,1,1,1,1    N=2 |
| 2,2,1,1,1    N=1 |
| 2,1,1,1,1,1  N=2 |
O------------------O

1234567 1234576 1235467 1235476
2314567 2314576 2315467 2315476
3124567 3124576 3125467 3125476

The cyclic group [3.02] on 123, pairs 45, 67 swap independently. Primary subgroups: 4.06 5.08 7.39; Normal subgroups: 2.01 3.02 4.06 4.07 5.08 7.39; Parent groups: 7.22 7.25 7.34; Normal supergroups: 7.20(48m) 7.22(24m) 7.25(24m) 7.34(24m).


O------------------O
| 3,2,1,1      N=2 |     [7.38]   Order 12m
| 3,1,1,1,1    N=2 |
| 2,2,2,1      N=3 |
| 2,2,1,1,1    N=3 |
| 2,1,1,1,1,1  N=1 |
O------------------O

1234567 1234576 1325467 1325476
2314567 2314576 2135467 2135476
3124567 3124576 3215467 3215476

The extent [3.01] on 123, 45 swap linked to parity of 123, 67 swap independently. Primary subgroups: 5.07 5.08 6.36 7.40; Normal subgroups: 2.01 3.02 5.07 5.08 7.40; Parent groups: 7.10 7.34; Normal supergroup: 7.34(24m).


O----------------O
| 3,2,2      N=2 |     [7.39]   Order 6p
| 3,1,1,1,1  N=2 |
| 2,2,1,1,1  N=1 |
O----------------O

1234567  1235476
2314567  2315476
3124567  3125476

The cyclic group [3.02] on 123, pairs 45, 67 swap in phase. Subgroups: 3.02 4.07; Normal subgroups: 3.02 4.07; Parent groups: 7.29 7.30 7.31 7.32 7.35 7.36 7.37; Normal supergroups: 7.20(48m) 7.22(24m) 7.23(24p) 7.24(24m) 7.25(24m) 7.26(24m) 7.27(24m) 7.29(12m) 7.30(12p) 7.31(12p) 7.32(12m) 7.34(24m) 7.35(12p) 7.36(12m) 7.37(12m).


O----------------O
| 3,1,1,1,1  N=2 |     [7.40]   Order 6m
| 2,2,2,1    N=3 |
O----------------O

1234567 2135476 2314567 3215476 3124567 1325476

The extent [3.01] on 123, pairs 45, 67 swap in phase linked to parity of 123. The rows of a Stedman Triples six. Subgroups: 3.02 6.37; Normal subgroup: 3.02; Parent groups: 7.32 7.36 7.38; Normal supergroups: 7.20(48m) 7.26(24m) 7.27(24m) 7.32(12m) 7.34(24m) 7.36(12m) 7.38(12m).


Logical sequence of Groups on 7 working bells

               +ve cycle sets                  -ve cycle sets
          7   5,  4,  3,  3,  3,  2,      6,  5,  4,  4,  3,  2,  2,
              1,  2,  3,  2,  1,  2,      1   2   3   1,  2,  2,  1,
              1   1   1   2   1,  1,                  1,  1,  2,  1,
                              1   1,                  1   1   1   1,
                              1   1                               1,
                                                                  1

7.01    720 504 630 280 210  70 105     840 504 420 210 420 105  21
7.02    720 504 630 280 210  70 105       -   -   -   -   -   -   -
7.03     48   -  42  56   -   -  21       -   -   -   -   -   -   -
7.04      6   -   -  14   -   -   -      14   -   -   -   -   7   -
7.05      6   -   -  14   -   -   -       -   -   -   -   -   -   -
7.06      6   -   -   -   -   -   -       -   -   -   -   -   7   -
7.07      6   -   -   -   -   -   -       -   -   -   -   -   -   -
7.08      -  24  30   -  20  20  25       -  24   -  30  40  15  11
7.09      -  24  30   -  20  20  25       -   -   -   -   -   -   -
7.10      -  24   -   -   -  20  15       -  24   -   -  20  15   1
7.11      -   4  10   -   -   -   5       -   4   -  10   -   5   1
7.12      -   4  10   -   -   -   5       -   -   -   -   -   -   -
7.13      -   4   -   -   -   -   5       -   4   -   -   -   5   1
7.14      -   4   -   -   -   -   -       -   4   -   -   -   -   1
7.15      -   4   -   -   -   -   -       -   -   -   -   -   5   -
7.16      -   -  18  16   6  10  21       -   -  12   6  36   9   9
7.19      -   -  18  16   6  10  21       -   -   -   -   -   -   -
7.28      -   -   6   8   -   -   9       -   -   -   -   -   -   -
7.20      -   -   6   -   6   2   9       -   -   4   2   4   9   5
7.23      -   -   6   -   6   2   9       -   -   -   -   -   -   -
7.26      -   -   6   -   2   2   1       -   -   4   2   -   3   3
7.25      -   -   6   -   2   2   1       -   -   -   -   4   6   2
7.30      -   -   6   -   2   2   1       -   -   -   -   -   -   -
7.17      -   -   -  16   6  10   3       -   -  12   6  12   -   6
7.18      -   -   -  16   6  10   3       -   -   -   -  24   9   3
7.21      -   -   -  16   6  10   3       -   -   -   -   -   -   -
7.33      -   -   -   8   -   -   3       -   -   -   -   -   -   -
7.22      -   -   -   -   6   2   3       -   -   4   2   4   -   2
7.27      -   -   -   -   6   2   3       -   -   -   -   -   9   3
7.31      -   -   -   -   6   2   3       -   -   -   -   -   -   -
7.24      -   -   -   -   2   2   7       -   -   4   2   -   6   -
7.34      -   -   -   -   2   2   7       -   -   -   -   4   3   5
7.35      -   -   -   -   2   2   7       -   -   -   -   -   -   -
7.29      -   -   -   -   2   2   1       -   -   4   2   -   -   -
7.37      -   -   -   -   2   2   1       -   -   -   -   4   -   2
7.32      -   -   -   -   2   2   1       -   -   -   -   -   6   -
7.36      -   -   -   -   2   2   1       -   -   -   -   -   3   3
7.39      -   -   -   -   2   2   1       -   -   -   -   -   -   -
7.38      -   -   -   -   -   2   3       -   -   -   -   2   3   1
7.40      -   -   -   -   -   2   -       -   -   -   -   -   3   -


Logical sequence of Groups on 6 working bells

              +ve cycle sets          -ve cycle sets
            5,  4,  3,  3,  2,      6   4,  3,  2,  2,
            1   2   3   1,  2,          1,  2,  2,  1,
                        1,  1,          1   1   2   1,
                        1   1                       1,
                                                    1

  6.01    144  90  40  40  45     120  90 120  15  15
  6.02    144  90  40  40  45       -   -   -   -   -
  6.03     24   -  20   -  15      20  30   -  10   -
  6.05     24   -  20   -  15       -   -   -   -   -
  6.04      -  18   4   4   9      12   -  12   6   6
  6.07      -  18   4   4   9       -   -   -   -   -
  6.06      -   6   8   -   9       8   6   -   7   3
  6.09      -   6   8   -   9       -   -   -   -   -
  6.17      -   6   -   8   9       -   6   8   3   7
  6.19      -   6   -   8   9       -   -   -   -   -
  6.20      -   2   -   -   5       -   2   -   3   3
  6.23      -   2   -   -   5       -   -   -   -   -
  6.22      -   2   -   -   1       -   2   -   1   1
  6.24      -   2   -   -   1       -   -   -   2   2
  6.26      -   2   -   -   1       -   -   -   -   -
  6.11      -   -   8   -   3       8   -   -   1   3
  6.10      -   -   8   -   3       -   6   -   6   -
  6.14      -   -   8   -   3       -   -   -   -   -
  6.08      -   -   4   4   9      12   -   -   6   -
  6.28      -   -   4   4   9       -   -  12   -   6
  6.30      -   -   4   4   9       -   -   -   -   -
  6.12      -   -   4   4   -       6   -   -   3   -
  6.29      -   -   4   4   -       -   -   6   -   3
  6.31      -   -   4   4   -       -   -   -   -   -
  6.13      -   -   2   -   3       2   -   -   4   -
  6.32      -   -   2   -   3       -   -   -   -   -
  6.15      -   -   2   -   -       2   -   -   1   -
  6.16      -   -   2   -   -       -   -   -   3   -
  6.33      -   -   2   -   -       -   -   -   -   -
  6.18      -   -   -   8   3       -   -   8   3   1
  6.25      -   -   -   -   3       -   2   -   2   -
  6.21      -   -   -   -   3       -   -   -   3   1
  6.34      -   -   -   -   3       -   -   -   1   3
  6.35      -   -   -   -   3       -   -   -   -   -
  6.27      -   -   -   -   1       -   -   -   2   -
  6.36      -   -   -   -   1       -   -   -   1   1
  6.37      -   -   -   -   -       -   -   -   1   -

The segregation of the cycle sets into +ve and -ve in this table, as in the other tables, emphasises the relationship between a mixed group and the group of half the order derived from it by selecting only the +ve perms. For example, both 6.08 and 6.28 are mixed and of order 36, while 6.30 has their +ve perms only, of order 18. All three groups have the same frequencies of +ve cycle sets.


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