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:
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----------01234 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------------01234 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----------01234 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----------01234 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----------01234 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-------------012345 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------------012345 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--------012345 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------------012345 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--------------012345 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---------------0123456 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-----------------0123456 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---------------0123456 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----------------0123456 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---------------0123456 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---------------0123456 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--------------0123456 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--------------0123456 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------------------0123456 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--------------0123456 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--------------0123456 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--------------0123456 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------------0123456 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-----------0123456 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----------------0123456 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----------------0123456 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--------------0123456 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----------------0123456 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----------------0123456 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--------------0123456 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--------------0123456 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----------------0123456 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--------------0123456 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--------------0123456 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--------------0123456 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----------0123456 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---------------0123456 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----------------0123456 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------------0123456 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-----------------01234567 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---------------O1234567 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-------------O1234567 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--------------O1234567 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--------O1234567 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-------------------O1234567 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-----------------O1234567 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------------------O1234567 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------------------O1234567 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---------------O1234567 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------------------O1234567 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----------------O1234567 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----------------O1234567 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------------------O1234567 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------------------O1234567 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------------------O1234567 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----------------O1234567 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----------------O1234567 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----------------O1234567 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----------------O1234567 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----------------O1234567 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----------------O1234567 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----------------O1234567 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------------------O1234567 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------------------O1234567 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------------------O1234567 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----------------O1234567 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----------------O1234567 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).
+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 -
+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.