3-bit Walsh permutation

There are A002884(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[edit]

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[edit]

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 A198380)
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 pattern The 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 A194602 (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

30
124

+

3

0

(1+1+1)

0

(1+1+1)

9

(3+3)

33

31

241

412

−

3

0

(1+1+1)

0

(1+1+1)

3

(2+2)

32 33

34

214		142
	421

+

4

1

(2+1+1)

1

(2+1+1)

14

(7)

41.a b 42.a b

40

	432							261
	432		641			413		261
341			641			413			416
341		452		Conjugacy classes: Inner circle 14_b Outer circle 14_a			251		416
		452					251

				243	612

				512	245

−

4

1

(2+1+1)

1

(2+1+1)

8

(4+2)

41.a b 45

43.a

	342			216
431					461

		215	542

−

4

1

(2+1+1)

1

(2+1+1)

8

(4+2)

41.a b 45

43.b

	621			423
152					254

		143	614

−

4

1

(2+1+1)

1

(2+1+1)

9

(3+3)

42.a b 46

44

	314			146
234					162

		425	521

+

4

1

(2+1+1)

1

(2+1+1)

3

(2+2)

45 46

47

	324			126
134					164

		125	524

−

5

2

(3+1+1)

3

(2+2+1)

3

(2+2)

50 53

51.a

217		742
	471

−

5

3

(2+2+1)

2

(3+1+1)

3

(2+2)

50 53

51.b

654		153
	623

+

5

2

(3+1+1)

3

(2+2+1)

3

(2+2)

50 54

52.a

127		724
	174

+

5

3

(2+2+1)

2

(3+1+1)

3

(2+2)

50 54

52.b

564		135
	326

+

5

2

(3+1+1)

3

(2+2+1)

8

(4+2)

53 57.m s

55.a

	741			417
472					271

		247	712

+

5

3

(2+2+1)

2

(3+1+1)

8

(4+2)

53 57.m s

55.b

	645			513
456					351

		263	632

−

5

2

(3+1+1)

3

(2+2+1)

9

(3+3)

54 57.m s

56.a

	721			427
172					274

		147	714

−

5

3

(2+2+1)

2

(3+1+1)

9

(3+3)

54 57.m s

56.b

	546			315
465					531

		236	362

∓

7

11

(3+2+2)

11

(3+2+2)

8

(4+2)

70 74.a b

72

657				753
657	567		735	753
	567		735
		376

		673

∓

7

11

(3+2+2)

11

(3+2+2)

14

(7)

71 74.a b

73

	375							576
	375		763			637		576
736			763			637			367
736		573		Conjugacy classes: Inner circle 14_b Outer circle 14_a			675		367
		573					675

				357	756

				765	537

−

5
6

3
5

(2+2+1)
(3+2+1)

3
5

(2+2+1)
(3+2+1)

14

(7)

BL.5 6 BM.5 6 BR.5 6

B

	372							276
	372		543			615		276
731			543			615			467
731		463		Conjugacy classes: Inner circle 14_a Outer circle 14_b			631		467
		463					631

				256	352

				715	547

+

5
6

3
5

(2+2+1)
(3+2+1)

3
5

(2+2+1)
(3+2+1)

9

(3+3)

BL.5 6 TL.5 6

CL

	732							267
	732		643			613		267
371			643			613			476
371		453					651		476
		453					651

				253	652

				517	745

+

5
6

3
5

(2+2+1)
(3+2+1)

3
5

(2+2+1)
(3+2+1)

8

(4+2)

BM.5 6 TM.5 6

CM

	726							327
	726		526			325		327
175			526			325			574
175		165					534		574
		165					534

				136	364

				137	764

+

5
6

3
5

(2+2+1)
(3+2+1)

3
5

(2+2+1)
(3+2+1)

14

(7)

BR.5 6 TR.5 6

CR

	672							273
	672		345			516		273
751			345			516			457
751		436		Conjugacy classes: Inner circle 14_b Outer circle 14_a			361		457
		436					361

				265	532

				713	647

−

5
6

3
5

(2+2+1)
(3+2+1)

3
5

(2+2+1)
(3+2+1)

9

(3+3)

TL.5 6 TM.5 6 TR.5 6

T.a

	762							237
	762		346			316		237
571			346			316			475
571		435					561		475
		435					561

				235	562

				317	746

−

5
6

3
5

(2+2+1)
(3+2+1)

3
5

(2+2+1)
(3+2+1)

9

(3+3)

TL.5 6 TM.5 6 TR.5 6

T.b

	627							723
	627		523			625		723
157			523			625			754
157		163					634		754
		163					634

				156	354

				173	674

Matrix patterns[edit]

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 matrices 5₂₃ 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 pattern The shade of the green corresponds to the number of collineations in the box.

Σ

E

permeq

mateq

3

30 34

32

124

421

3

31 34

33

142	241
412	214

4

43.a b 40

41.a

143	341		542	245
416	614		512	215

4

43.a b 40

41.b

342	243		152	251
612	216		452	254

4

40 44

42.a

146	641
413	314

4

40 44

42.b

162	261
432	234

4

43.a b 47

45

164	461		126	621
431	134		423	324

4

44 47

46

521	125
524	425

5₂₃

51.a b 52.a b

50

174

471

326

623

5₂₃

51.a b 55.a b

53

712	217		351	153
742	247		654	456

5₂₃

52.a b 56.a b

54

724	427		135	531
721	127		465	564

5₂₃

55.a b 56.a b

57.m

362	263		172	271
632	236		472	274

5₂₃

55.a b 56.a b

57.s

714	417		315	513
741	147		645	546

7

72

70

673

376

7

73

71

736	637		375	573
763	367		675	576

7

72 73

74.a

753	357
756	657

7

72 73

74.b

735	537
765	567

5₃₃

B CL

BL.5

352	253
652	256

6

B CL

BL.6

715	517
745	547

5₃₃

B CM

BM.5

631	136
364	463

6

B CM

BM.6

731	137
764	467

5₃₃

B CR

BR.5

615	516
345	543

6

B CR

BR.6

372	273
672	276

5₃₃

CL T.a b

TL.5

346	643		156	651
613	316		453	354

6

CL T.a b

TL.6

732	237		371	173
762	267		674	476

5₃₃

CM T.a b

TM.5

165	561		526	625
435	534		523	325

6

CM T.a b

TM.6

726	627		175	571
723	327		475	574

5₃₃

CR T.a b

TR.5

562	265		163	361
532	235		436	634

6

CR T.a b

TR.6

713	317		751	157
746	647		754	457

Graphs[edit]

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.