# 3-bit Walsh permutation/arrow patterns

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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

+ middle S3

+ 3
6
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+ middle S4

+ 4 22 121
3

+ middle S5a F121

+ 5a 22 121
3

+ middle S5a F132

+ 5a 22 132
6 * 2

outside: + middle S6 inside: + middle S5b

+ 5b
6
24 1
2

+ sides S3

+ 3 33 3
6

+ sides S5a F2

+ 5a 24 2
6

+ sides S5a F3

+ 5a 24 3
6 * 2

outside: + sides S4 C7b inside: + sides S4 C7a

+ 4 7
6 * 2

outside: + sides S5b C33 inside: + sides S6 C33

+ 5b
6
33 2
6 * 2

outside: + sides S6 C7a inside: + sides S5b C7b

+ 5b
6
7
6 * 2

outside: − matrix triangles inside: + matrix triangles

± 7 7
3 * 2

outside: − matrix diagonal inside: + matrix diagonal

± 7 24 2
3

S3

3 22 132
3

S5a C22 F222

5a 22 222
3

S5a C22 F132

5a 22 132
6

S5a C33 F1

5a 33 1
6

S5a C33 F3

5a 33 3
6

S4 C33

4 33 1
6

S4 C24 F2

4 24 2
6

S4 C24 F1

4 24 1
6 * 2

outside: S6 C33 F2 inside: S5b C33 F2

5b
6
33 2
6 * 2

outside: S5b C33 F1 inside: S6 C33 F1

5b
6
33 1
6 * 2

outside: S5b C7a inside: S6 C7b

5b
6
7