3-bit Walsh permutation/cycle shapes

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matrix sums from 3 to 6
fixed points

This table shows the 24 arrow patterns of the 168 3-bit Walsh permutations (as Fano plane collineations).

The typical shape of a collection is a hexagon, showing the two rotations and three reflections of some initial pattern.
The one pattern with rotational symmetry is shown as a pair. Patterns with mirror symmetry are shown as triangles.
Where the hexagon or triangle does not contain the inverse permutations, two nested polygons are shown.
These are the types of collections shown below:

  • neutral permutation
  • 1 pair (of rotations)

  • 5 single triangles
  • 1 double triangle

  • 8 single hexagons
  • 8 double hexagons

In the column graph, the shown permutations are highlighted in the respective component of the neighbor graph. All except 9 are in the cube-like clusters with 25 vertices. Apart from the center, there are four types of vertices, which are marked with suits.

Each permutation has a set of fixed points.

  • The neutral permutation has only fixed points.
  • 7-cycles have none.

  • For cycle type 2+2 they form a line of the Fano plane.

  • Cycle types 2+4 and 3+3 have single fixed points.

The points can be characterized by their binary weight, and the lines by the weights of their points. This property is in the column fixed.

size collection graph det sum cycles fixed
&1 1

Walsh permutation 124 Fano.svg


Fano patterns in neighbor graph; positive middle star (124).svg

&1 + 3 &0 &1
&5 6

Walsh permutation 134 Fano.svg


Walsh permutation 164 Fano.svg


Walsh permutation 324 Fano.svg


Walsh permutation 126 Fano.svg


Walsh permutation 524 Fano.svg


Walsh permutation 125 Fano.svg



London Fano patterns in neighbor graph; positive middle spades (134).svg

&1 + 4 &2 22 &2 121
&3 3

Walsh permutation 127 Fano.svg


Walsh permutation 724 Fano.svg


Walsh permutation 174 Fano.svg



Rome Fano patterns in neighbor graph; positive middle hearts 127.svg

&1 + 5a &2 22 &2 121
&3 3

Walsh permutation 564 Fano.svg


Walsh permutation 135 Fano.svg


Walsh permutation 326 Fano.svg



Florence Fano patterns in neighbor graph; positive middle hearts 564.svg

&1 + 5a &2 22 &3 132
&6 6 * 2

Walsh permutation 137 Fano.svg


Walsh permutation 764 Fano.svg


Walsh permutation 136 Fano.svg


Walsh permutation 364 Fano.svg


Walsh permutation 327 Fano.svg


Walsh permutation 726 Fano.svg


Walsh permutation 325 Fano.svg


Walsh permutation 526 Fano.svg


Walsh permutation 534 Fano.svg


Walsh permutation 165 Fano.svg


Walsh permutation 574 Fano.svg


Walsh permutation 175 Fano.svg



Lima inside: 5 Fano patterns in neighbor graph; positive middle clubs (136).svg outside: 6 Fano patterns in neighbor graph; positive middle diamonds (137).svg

&1 + 5b
6
&3 24 &5 1
&2 2

Walsh permutation 241 Fano.svg


Walsh permutation 412 Fano.svg


Tokyo Fano patterns in neighbor graph; positive sides star (241).svg

&1 + 3 &4 33 &7 3
&5 6

Walsh permutation 741 Fano.svg


Walsh permutation 417 Fano.svg


Walsh permutation 472 Fano.svg


Walsh permutation 271 Fano.svg


Walsh permutation 247 Fano.svg


Walsh permutation 712 Fano.svg



New York Fano patterns in neighbor graph; positive sides hearts 741.svg

&1 + 5a &3 24 &6 2
&5 6

Walsh permutation 263 Fano.svg


Walsh permutation 632 Fano.svg


Walsh permutation 513 Fano.svg


Walsh permutation 645 Fano.svg


Walsh permutation 351 Fano.svg


Walsh permutation 456 Fano.svg



San Francisco Fano patterns in neighbor graph; positive sides hearts 263.svg

&1 + 5a &3 24 &7 3
&6 6 * 2

Walsh permutation 413 Fano.svg


Walsh permutation 641 Fano.svg


Walsh permutation 261 Fano.svg


Walsh permutation 432 Fano.svg


Walsh permutation 243 Fano.svg


Walsh permutation 612 Fano.svg


Walsh permutation 512 Fano.svg


Walsh permutation 245 Fano.svg


Walsh permutation 341 Fano.svg


Walsh permutation 416 Fano.svg


Walsh permutation 452 Fano.svg


Walsh permutation 251 Fano.svg



Cairo inside: a Fano patterns in neighbor graph; positive sides spades yellow (261).svg outside: b Fano patterns in neighbor graph; positive sides spades orange (413).svg

&1 + 4 &5 7 &8
&6 6 * 2

Walsh permutation 253 Fano.svg


Walsh permutation 652 Fano.svg


Walsh permutation 517 Fano.svg


Walsh permutation 745 Fano.svg


Walsh permutation 613 Fano.svg


Walsh permutation 643 Fano.svg


Walsh permutation 267 Fano.svg


Walsh permutation 732 Fano.svg


Walsh permutation 476 Fano.svg


Walsh permutation 371 Fano.svg


Walsh permutation 651 Fano.svg


Walsh permutation 453 Fano.svg



Karachi inside: 6 Fano patterns in neighbor graph; positive sides diamonds white (517).svg outside: 5 Fano patterns in neighbor graph; positive sides clubs white (253).svg

&1 + 5b
6
&4 33 &6 2
&6 6 * 2

Walsh permutation 713 Fano.svg


Walsh permutation 647 Fano.svg


Walsh permutation 265 Fano.svg


Walsh permutation 532 Fano.svg


Walsh permutation 273 Fano.svg


Walsh permutation 672 Fano.svg


Walsh permutation 516 Fano.svg


Walsh permutation 345 Fano.svg


Walsh permutation 361 Fano.svg


Walsh permutation 436 Fano.svg


Walsh permutation 457 Fano.svg


Walsh permutation 751 Fano.svg



Alexandria inside: b (5b) Fano patterns in neighbor graph; positive sides clubs orange (265).svg outside: a (6) Fano patterns in neighbor graph; positive sides diamonds yellow (713).svg

&1 + 5b
6
&5 7 &8
&6 6 * 2

Walsh permutation 763 Fano.svg


Walsh permutation 637 Fano.svg


Walsh permutation 375 Fano.svg


Walsh permutation 576 Fano.svg


Walsh permutation 573 Fano.svg


Walsh permutation 675 Fano.svg


Walsh permutation 736 Fano.svg


Walsh permutation 367 Fano.svg


Walsh permutation 765 Fano.svg


Walsh permutation 537 Fano.svg


Walsh permutation 357 Fano.svg


Walsh permutation 756 Fano.svg



Kinshasa inside: a (+) Fano patterns in neighbor graph; positive matrix triangles (375).svg outside: b (−) Fano patterns in neighbor graph; negative matrix triangles (763).svg

&2 ± 7 &5 7 &8
&4 3 * 2

Walsh permutation 567 Fano.svg


Walsh permutation 735 Fano.svg


Walsh permutation 657 Fano.svg


Walsh permutation 753 Fano.svg


Walsh permutation 673 Fano.svg


Walsh permutation 376 Fano.svg


Rio inside: + Fano patterns in neighbor graph; positive matrix diagonal (657).svg outside: Fano patterns in neighbor graph; negative matrix diagonal (567).svg

&2 ± 7 &3 24 &6 2
&3 3

Walsh permutation 214 Fano.svg


Walsh permutation 142 Fano.svg


Walsh permutation 421 Fano.svg



Paris Fano patterns in neighbor graph; negative star (214).svg

&3 3 &2 22 &3 132
&3 3

Walsh permutation 217 Fano.svg


Walsh permutation 742 Fano.svg


Walsh permutation 471 Fano.svg



Berlin Fano patterns in neighbor graph; negative hearts red 217.svg

&3 5a &2 22 &4 222
&3 3

Walsh permutation 654 Fano.svg


Walsh permutation 153 Fano.svg


Walsh permutation 623 Fano.svg



Hamburg Fano patterns in neighbor graph; negative hearts red 654.svg

&3 5a &2 22 &3 132
&5 6

Walsh permutation 274 Fano.svg


Walsh permutation 172 Fano.svg


Walsh permutation 714 Fano.svg


Walsh permutation 147 Fano.svg


Walsh permutation 721 Fano.svg


Walsh permutation 427 Fano.svg



Beijing Fano patterns in neighbor graph; negative hearts white 274.svg

&3 5a &4 33 &5 1
&5 6

Walsh permutation 236 Fano.svg


Walsh permutation 362 Fano.svg


Walsh permutation 315 Fano.svg


Walsh permutation 546 Fano.svg


Walsh permutation 531 Fano.svg


Walsh permutation 465 Fano.svg



Shanghai Fano patterns in neighbor graph; negative hearts white 236.svg

&3 5a &4 33 &7 3
&5 6

Walsh permutation 234 Fano.svg


Walsh permutation 162 Fano.svg


Walsh permutation 314 Fano.svg


Walsh permutation 146 Fano.svg


Walsh permutation 521 Fano.svg


Walsh permutation 425 Fano.svg



Seoul Fano patterns in neighbor graph; negative spades white (234).svg

&3 4 &4 33 &5 1
&5 6

Walsh permutation 215 Fano.svg


Walsh permutation 542 Fano.svg


Walsh permutation 216 Fano.svg


Walsh permutation 342 Fano.svg


Walsh permutation 461 Fano.svg


Walsh permutation 431 Fano.svg



Toronto Fano patterns in neighbor graph; negative spades green 215.svg

&3 4 &3 24 &6 2
&5 6

Walsh permutation 254 Fano.svg


Walsh permutation 152 Fano.svg


Walsh permutation 614 Fano.svg


Walsh permutation 143 Fano.svg


Walsh permutation 621 Fano.svg


Walsh permutation 423 Fano.svg



Montreal Fano patterns in neighbor graph; negative spades green 254.svg

&3 4 &3 24 &5 1
&6 6 * 2

Walsh permutation 237 Fano.svg


Walsh permutation 762 Fano.svg


Walsh permutation 316 Fano.svg


Walsh permutation 346 Fano.svg


Walsh permutation 317 Fano.svg


Walsh permutation 746 Fano.svg


Walsh permutation 235 Fano.svg


Walsh permutation 562 Fano.svg


Walsh permutation 435 Fano.svg


Walsh permutation 561 Fano.svg


Walsh permutation 571 Fano.svg


Walsh permutation 475 Fano.svg



Delhi inside: 5 Fano patterns in neighbor graph; negative clubs white 316.svg outside: 6 Fano patterns in neighbor graph; negative diamonds white 237.svg

&3 5b
6
&4 33 &6 2
&6 6 * 2

Walsh permutation 163 Fano.svg


Walsh permutation 634 Fano.svg


Walsh permutation 157 Fano.svg


Walsh permutation 754 Fano.svg


Walsh permutation 523 Fano.svg


Walsh permutation 625 Fano.svg


Walsh permutation 627 Fano.svg


Walsh permutation 723 Fano.svg


Walsh permutation 674 Fano.svg


Walsh permutation 173 Fano.svg


Walsh permutation 354 Fano.svg


Walsh permutation 156 Fano.svg



Mumbai inside: 6 Fano patterns in neighbor graph; negative diamonds white 157.svg outside: 5 Fano patterns in neighbor graph; negative clubs white 163.svg

&3 5b
6
&4 33 &5 1
&6 6 * 2

Walsh permutation 463 Fano.svg


Walsh permutation 631 Fano.svg


Walsh permutation 731 Fano.svg


Walsh permutation 467 Fano.svg


Walsh permutation 543 Fano.svg


Walsh permutation 615 Fano.svg


Walsh permutation 372 Fano.svg


Walsh permutation 276 Fano.svg


Walsh permutation 547 Fano.svg


Walsh permutation 715 Fano.svg


Walsh permutation 352 Fano.svg


Walsh permutation 256 Fano.svg



Lagos inside: b (6) Fano patterns in neighbor graph; negative diamonds orange (731).svg outside: a (5b) Fano patterns in neighbor graph; negative clubs yellow (463).svg

&3 5b
6
&5 7 &8