# 3-bit Walsh permutation

There are (3) = (8-1)*(8-2)*(8-4) = 168 Walsh permutations of 8 elements.
The compression matrices are the elements of the general linear group GL(3,2)
and the permutations correspond to the elements of PSL(2,7), the symmetry group of the Fano plane.

A table of all 168 can be found here.

## Fano plane collineations

Fano plane with nimber labels

The Walsh permutations leave the 0 unchanged,
so they can be interpreted as permutations of the seven points of the Fano plane
and it happens that these are the Fano plane's collineations.

E.g. these

 124 364 524 764

are the four different powers of the 3-bit Gray code permutation,
and this

 421

is the 3-bit bit-reversal permutation.

### Permutation patterns

The blue table shows 24 equivalence classes
containing turned, reflected and complemented collineations.
These are it's columns:

 D Real determinant of the 3×3 compression matrices (+ stands for 1, − for −1)In two rows ∓ is shown: Determinants in the inner circle are −, those in the outer circle are + Σ Number of ones in the compression matrices (sorting Co Ro will sort also this column)In six rows 5 and 6 are shown: Sums in the inner circle are 5, those in the outer circle are 6 (analogously for Co Ro) Co Ro Partitions representing the number of ones in the columns and rows of the compression matrices Cy Partition representing the cycle structure of the permutations (compare ) F Fixed points (images that are not rotationally symmetric are examples) E Example permutation mateq Matrix patterns that appear in the blue box (links go to the green table) permeq Collineations ordered by permutation patternThe shade of the blue corresponds to the number of collineations in the box (1,2,3,6,12).

The violet numbers represent partitions by index numbers of (compare this table).
As these numbers are everything but intuitive, the actual partitions are always shown in parentheses.
The sums shown in Co Ro give the corresponding entry in column Σ. In Cy only the non-one addends are shown.

D  Σ  Co Ro Cy F E mateq permeq
+   3      0 (1+1+1)  0 (1+1+1)      0         32
+ 3  0 (1+1+1)  0 (1+1+1)  9 (3+3) 33
3  0 (1+1+1)  0 (1+1+1)  3 (2+2) 32 33
+ 4  1 (2+1+1)  1 (2+1+1) 14 (7) 41.ab 42.ab
4  1 (2+1+1)  1 (2+1+1)  8 (4+2) 41.ab 45
4  1 (2+1+1)  1 (2+1+1)  8 (4+2) 41.ab 45
4  1 (2+1+1)  1 (2+1+1)  9 (3+3) 42.ab 46
+ 4  1 (2+1+1)  1 (2+1+1)  3 (2+2) 45 46
5  2 (3+1+1)  3 (2+2+1)  3 (2+2) 50 53
5  3 (2+2+1)  2 (3+1+1)  3 (2+2) 50 53
+ 5  2 (3+1+1)  3 (2+2+1)  3 (2+2) 50 54
+ 5  3 (2+2+1)  2 (3+1+1)  3 (2+2) 50 54
+ 5  2 (3+1+1)  3 (2+2+1)  8 (4+2)  53 57.ms
+ 5  3 (2+2+1)  2 (3+1+1)  8 (4+2)  53 57.ms
5  2 (3+1+1)  3 (2+2+1)  9 (3+3)  54 57.ms
5  3 (2+2+1)  2 (3+1+1)  9 (3+3)  54 57.ms
7 11 (3+2+2) 11 (3+2+2)  8 (4+2) 70 74.ab
7 11 (3+2+2) 11 (3+2+2) 14 (7) 71 74.ab
5
6
3
5
(2+2+1)
(3+2+1)
3
5
(2+2+1)
(3+2+1)
14 (7) BL.56 BM.56 BR.56
+ 5
6
3
5
(2+2+1)
(3+2+1)
3
5
(2+2+1)
(3+2+1)
9 (3+3) BL.56 TL.56
+ 5
6
3
5
(2+2+1)
(3+2+1)
3
5
(2+2+1)
(3+2+1)
8 (4+2) BM.56 TM.56
+ 5
6
3
5
(2+2+1)
(3+2+1)
3
5
(2+2+1)
(3+2+1)
14 (7) BR.56 TR.56
5
6
3
5
(2+2+1)
(3+2+1)
3
5
(2+2+1)
(3+2+1)
9 (3+3) TL.56 TM.56 TR.56
5
6
3
5
(2+2+1)
(3+2+1)
3
5
(2+2+1)
(3+2+1)
9 (3+3) TL.56 TM.56 TR.56

### Matrix patterns

The green table shows 29 equivalence classes
containing collineations with turned and reflected compression matrices.
These are it's columns:

 Σ Number of ones in the compression matrices5₂₃ tells that the matrix has a row or column with only ones, compare CoRo in blue table E Example matrix permeq Permutation patterns that appear in the green box (links go to the blue table) mateq Collineations ordered by matrix patternThe shade of the green corresponds to the number of collineations in the box.
Σ   E   permeq mateq
3

30 34
3

31 34
4

43.ab 40
4

43.ab 40
4

40 44
4

40 44
4

43.ab 47
4

44 47
5₂₃

51.ab 52.ab
5₂₃

51.ab 55.ab
5₂₃

52.ab 56.ab
5₂₃

55.ab 56.ab
5₂₃

55.ab 56.ab
7

72
7

73
7

72 73
7

72 73
5₃₃

B CL
6

B CL
5₃₃

B CM
6

B CM
5₃₃

B CR
6

B CR
5₃₃

CL T.ab
6

CL T.ab
5₃₃

CM T.ab
6

CM T.ab
5₃₃

CR T.ab
6

CR T.ab

### Graphs

The connections between permutation patterns (blue) and matrix patterns (green) can be visualized in five graphs.
It can be seen that, apart from the graph on the right, all compression matrices in the same graph have the same number of ones,
used as the first digit of the node numbers.

 The compression matices have 3 ones. The compression matices have 7 ones. The compression matices have 5 or 6 ones.The nodes are named after their positions:T, C, B stand for top, center, bottom,L, M, R stand for left, middle, right. The compression matices have 4 ones. The compression matices have 5 ones.

## Conjugacy classes

The cycle structures 0, 3, 8 and 9 uniquely define four conjugacy classes with 1, 21, 42 and 56 elements.
The 48 permutations with cycle structure 14 (the 7-cycle) form two distinct conjugacy classes with 24 elements,
here called 14a and 14b. The inverse of a permutation in one of these is in the other one,
so in the blue table above each of the four boxes in a row with a 14 contains six permutations from 14a and six from 14b.

When the elements x and y belong to the conjugacy class C their powers xn and yn belong to the conjugacy class Cn.
When the conjugacy class is defined by the cycle structure it's powers follow from the cycle structure.
The powers of 14a and 14b can be seen in the examples below:

Powers of a permutation with two 3-cycles:

 wp(7,2,1)1 wp(7,2,1)2 721 427 9 9 56.a 56.a 54 54

Powers of a permutation with a 4-cycle and a 2-cycle:

 wp(6,2,1)1 wp(6,2,1)2 wp(6,2,1)3 621 326 423 8 3 8 43.b 52.b 43.b 45 50 45

Powers of 14a and 14b:

 wp(7,6,5)1 wp(7,6,5)2 wp(7,6,5)3 wp(7,6,5)4 wp(7,6,5)5 wp(7,6,5)6 765 432 516 273 641 357 14a 14a 14b 14a 14b 14b 73 40 CR CR 40 73 74.b 42.b BR.5 BR.6 42.a 74.a
 wp(7,3,1)1 wp(7,3,1)2 wp(7,3,1)3 wp(7,3,1)4 wp(7,3,1)5 wp(7,3,1)6 731 547 615 276 352 463 14b 14b 14a 14b 14a 14a B B B B B B BM.6 BL.6 BR.5 BR.6 BL.5 BM.5

## Cayley table

In the following Cayley table the conjugacy classes of the permutations are shown by colors.

 The Walsh functions 124 , 421 , 376 , 241 , 765 , 357 belong to the conjugacy classes 0 , 3 (2+2) , 8 (4+2) , 9 (3+3) , 14a (7) , 14b (7) with 1 , 21 , 42 , 56 , 24 , 24 elements.

The Cayley table can be found here.