# 3-bit Walsh permutation/cycle shapes

 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

+ 3
6

London

+ 4 22 121
3

Rome

+ 5a 22 121
3

Florence

+ 5a 22 132
6 * 2

Lima inside: 5 outside: 6

+ 5b
6
24 1
2

Tokyo

+ 3 33 3
6

New York

+ 5a 24 2
6

San Francisco

+ 5a 24 3
6 * 2

Cairo inside: a outside: b

+ 4 7
6 * 2

Karachi inside: 6 outside: 5

+ 5b
6
33 2
6 * 2

Alexandria inside: b (5b) outside: a (6)

+ 5b
6
7
6 * 2

Kinshasa inside: a (+) outside: b (−)

± 7 7
3 * 2

Rio inside: + outside:

± 7 24 2
3

Paris

3 22 132
3

Berlin

5a 22 222
3

Hamburg

5a 22 132
6

Beijing

5a 33 1
6

Shanghai

5a 33 3
6

Seoul

4 33 1
6

Toronto

4 24 2
6

Montreal

4 24 1
6 * 2

Delhi inside: 5 outside: 6

5b
6
33 2
6 * 2

Mumbai inside: 6 outside: 5

5b
6
33 1
6 * 2

Lagos inside: b (6) outside: a (5b)

5b
6
7