# 3-bit Walsh permutation; table

Array of 2-element subsets of an 8-element set
(compare as a square array)

The sortable table shows all 168 3-bit Walsh permutations. These are it's columns:

 CV Compression vector CM Compression matrix D Real determinant of the compression matrix (+ stands for 1, − for −1) Σ Number of ones in the compression matrix # Index number of the finite permutation (compare ) Permutation Walsh permutations of numbers 0,...,7 Inversion Vector Inversion vector (reflected factorial number) showing the number of bigger elements to the left of each element IN Inversion number (sum of the inversion vector) IS Shows which 2-cycles are part of the inversion set. Compare file on the right. PM Permutation matrix (to permute rows, e.g. the elements of a column vector) FPC Fano plane collineation Cy Partition representing the cycle structure of the permutation (compare )14a and 14b are conjugacy classes (see here), so the indexes are information beyond the cycle structure. I Sorting this column will show inverse permutations next to each other.

Plain text tables: Compression vectors, Permutations

Code of the binary matrices can be copied from their gray and red representations.

CV   CM   D Σ # Permutation
Inversion Vector
IN IS PM FPC Cy I
1 2 4

+ 3 0 0 1 2 3   4 5 6 7
0 0 0 0   0 0 0 0
0

0
7 2 4

+ 5 126 0 1 3 2   5 4 6 7
0 0 0 1   0 1 0 0
2

3   (2+2)>
1 4 2

3 288 0 1 4 5   2 3 6 7
0 0 0 0   2 2 0 0
4

3   (2+2)>
7 4 2

5 414 0 1 5 4   3 2 6 7
0 0 0 1   2 3 0 0
6

3   (2+2)>
2 1 4

3 722 0 2 1 3   4 6 5 7
0 0 1 0   0 0 1 0
2

3   (2+2)>
2 7 4

5 852 0 2 3 1   6 4 5 7
0 0 0 2   0 1 1 0
4

9   (3+3)>
4 1 2

+ 3 1032 0 2 4 6   1 3 5 7
0 0 0 0   3 2 1 0
6

9   (3+3)>
4 7 2

+ 5 1254 0 2 6 4   3 1 5 7
0 0 0 1   2 4 1 0
8

8   (4+2)>
7 1 4

5 1448 0 3 1 2   5 6 4 7
0 0 1 1   0 0 2 0
4

9   (3+3)>
1 7 4

+ 5 1574 0 3 2 1   6 5 4 7
0 0 1 2   0 1 2 0
6

3   (2+2)>
7 1 2

+ 5 1872 0 3 5 6   1 2 4 7
0 0 0 0   3 3 2 0
8

8   (4+2)>
1 7 2

5 1998 0 3 6 5   2 1 4 7
0 0 0 1   3 4 2 0
10

9   (3+3)>
2 4 1

+ 3 2210 0 4 1 5   2 6 3 7
0 0 1 0   2 0 3 0
6

9   (3+3)>
4 2 1

3 2354 0 4 2 6   1 5 3 7
0 0 1 0   3 1 3 0
8

3   (2+2)>
2 4 7

+ 5 2532 0 4 5 1   6 2 3 7
0 0 0 2   0 3 3 0
8

8   (4+2)>
4 2 7

5 2676 0 4 6 2   5 1 3 7
0 0 0 2   1 4 3 0
10

9   (3+3)>
7 4 1

+ 5 2936 0 5 1 4   3 6 2 7
0 0 1 1   2 0 4 0
8

8   (4+2)>
7 2 1

5 3194 0 5 3 6   1 4 2 7
0 0 1 0   3 2 4 0
10

9   (3+3)>
1 4 7

5 3254 0 5 4 1   6 3 2 7
0 0 1 2   0 3 4 0
10

9   (3+3)>
1 2 7

+ 5 3420 0 5 6 3   4 1 2 7
0 0 0 2   2 4 4 0
12

3   (2+2)>
4 7 1

5 3776 0 6 2 4   3 5 1 7
0 0 1 1   2 1 5 0
10

3   (2+2)>
2 7 1

+ 5 3920 0 6 3 5   2 4 1 7
0 0 1 1   3 2 5 0
12

8   (4+2)>
4 1 7

+ 5 3998 0 6 4 2   5 3 1 7
0 0 1 2   1 3 5 0
12

8   (4+2)>
2 1 7

5 4142 0 6 5 3   4 2 1 7
0 0 1 2   2 4 5 0
14

3   (2+2)>
3 2 4

+ 4 5046 0 1 3 2   4 5 7 6
0 0 0 1   0 0 0 1
2

3   (2+2)>
5 2 4

+ 4 5160 0 1 2 3   5 4 7 6
0 0 0 0   0 1 0 1
2

3   (2+2)>
3 4 2

4 5334 0 1 5 4   2 3 7 6
0 0 0 1   2 2 0 1
6

8   (4+2)>
5 4 2

4 5448 0 1 4 5   3 2 7 6
0 0 0 0   2 3 0 1
6

8   (4+2)>
3 1 4

4 5768 0 3 1 2   4 7 5 6
0 0 1 1   0 0 1 1
4

9   (3+3)>
5 7 4

+ 6 5894 0 3 2 1   7 4 5 6
0 0 1 2   0 1 1 1
6

8   (4+2)>
5 1 2

+ 4 6192 0 3 4 7   1 2 5 6
0 0 0 0   3 3 1 1
8

14a   (7)>
3 7 2

6 6318 0 3 7 4   2 1 5 6
0 0 0 1   3 4 1 1
10

14b   (7)>
6 1 4

4 6482 0 2 1 3   5 7 4 6
0 0 1 0   0 0 2 1
4

8   (4+2)>
6 7 4

6 6612 0 2 3 1   7 5 4 6
0 0 0 2   0 1 2 1
6

9   (3+3)>
6 1 2

+ 4 6792 0 2 5 7   1 3 4 6
0 0 0 0   3 2 2 1
8

14b   (7)>
6 7 2

+ 6 7014 0 2 7 5   3 1 4 6
0 0 0 1   2 4 2 1
10

14a   (7)>
3 4 1

+ 4 7256 0 5 1 4   2 7 3 6
0 0 1 1   2 0 3 1
8

14a   (7)>
5 2 1

4 7514 0 5 2 7   1 4 3 6
0 0 1 0   3 2 3 1
10

9   (3+3)>
5 4 7

6 7574 0 5 4 1   7 2 3 6
0 0 1 2   0 3 3 1
10

14b   (7)>
3 2 7

+ 6 7740 0 5 7 2   4 1 3 6
0 0 0 2   2 4 3 1
12

8   (4+2)>
6 4 1

+ 4 7970 0 4 1 5   3 7 2 6
0 0 1 0   2 0 4 1
8

14b   (7)>
6 2 1

4 8114 0 4 3 7   1 5 2 6
0 0 1 0   3 1 4 1
10

8   (4+2)>
6 4 7

+ 6 8292 0 4 5 1   7 3 2 6
0 0 0 2   0 3 4 1
10

14a   (7)>
6 2 7

6 8436 0 4 7 3   5 1 2 6
0 0 0 2   1 4 4 1
12

9   (3+3)>
3 7 1

+ 6 8840 0 7 3 4   2 5 1 6
0 0 1 1   3 1 5 1
12

9   (3+3)>
5 7 1

6 8936 0 7 2 5   3 4 1 6
0 0 1 1   2 2 5 1
12

9   (3+3)>
3 1 7

6 9062 0 7 5 2   4 3 1 6
0 0 1 2   2 3 5 1
14

9   (3+3)>
5 1 7

+ 6 9158 0 7 4 3   5 2 1 6
0 0 1 2   1 4 5 1
14

9   (3+3)>
2 3 4

4 10092 0 2 3 1   4 6 7 5
0 0 0 2   0 0 0 2
4

9   (3+3)>
2 5 4

4 10202 0 2 1 3   6 4 7 5
0 0 1 0   0 1 0 2
4

8   (4+2)>
4 3 2

+ 4 10398 0 2 6 4   1 3 7 5
0 0 0 1   3 2 0 2
8

14a   (7)>
4 5 2

+ 4 10608 0 2 4 6   3 1 7 5
0 0 0 0   2 4 0 2
8

14b   (7)>
1 3 4

+ 4 10814 0 3 2 1   4 7 6 5
0 0 1 2   0 0 1 2
6

3   (2+2)>
7 5 4

6 10928 0 3 1 2   7 4 6 5
0 0 1 1   0 1 1 2
6

9   (3+3)>
7 3 2

+ 6 11238 0 3 7 4   1 2 6 5
0 0 0 1   3 3 1 2
10

9   (3+3)>
1 5 2

4 11352 0 3 4 7   2 1 6 5
0 0 0 0   3 4 1 2
10

8   (4+2)>
1 6 4

+ 4 11520 0 1 2 3   6 7 4 5
0 0 0 0   0 0 2 2
4

3   (2+2)>
7 6 4

+ 6 11646 0 1 3 2   7 6 4 5
0 0 0 1   0 1 2 2
6

8   (4+2)>
1 6 2

4 11808 0 1 6 7   2 3 4 5
0 0 0 0   2 2 2 2
8

9   (3+3)>
7 6 2

6 11934 0 1 7 6   3 2 4 5
0 0 0 1   2 3 2 2
10

9   (3+3)>
4 3 1

4 12320 0 6 2 4   1 7 3 5
0 0 1 1   3 0 3 2
10

8   (4+2)>
2 5 1

+ 4 12530 0 6 1 7   2 4 3 5
0 0 1 0   2 2 3 2
10

14b   (7)>
2 3 7

6 12660 0 6 7 1   4 2 3 5
0 0 0 2   2 3 3 2
12