3-bit Walsh permutation

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

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  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
&1 +		3		0	(1+1+1)	0	(1+1+1)	&1		0			&8				32		30 124
&1 +		3		0	(1+1+1)	0	(1+1+1)	&1		9	(3+3)		&4				33		31 241 412
&3 −		3		0	(1+1+1)	0	(1+1+1)	&1		3	(2+2)		&6				32 33		34 214 142 421
&1 +		4		1	(2+1+1)	1	(2+1+1)	&2		14	(7)		&1				41.ab 42.ab		40 432 261 641 413 341 416 452 Conjugacy classes: Inner circle 14b Outer circle 14a 251 243 612 512 245
&3 −		4		1	(2+1+1)	1	(2+1+1)	&2		8	(4+2)		&2				41.ab 45		43.a 342 216 431 461 215 542
&3 −		4		1	(2+1+1)	1	(2+1+1)	&2		8	(4+2)		&3				41.ab 45		43.b 621 423 152 254 143 614
&3 −		4		1	(2+1+1)	1	(2+1+1)	&2		9	(3+3)		&2				42.ab 46		44 314 146 234 162 425 521
&1 +		4		1	(2+1+1)	1	(2+1+1)	&2		3	(2+2)		&5				45 46		47 324 126 134 164 125 524
&3 −		5		2	(3+1+1)	3	(2+2+1)	&3		3	(2+2)		&6				50 53		51.a 217 742 471
&3 −		5		3	(2+2+1)	2	(3+1+1)	&4		3	(2+2)		&7				50 53		51.b 654 153 623
&1 +		5		2	(3+1+1)	3	(2+2+1)	&3		3	(2+2)		&6				50 54		52.a 127 724 174
&1 +		5		3	(2+2+1)	2	(3+1+1)	&4		3	(2+2)		&5				50 54		52.b 564 135 326
&1 +		5		2	(3+1+1)	3	(2+2+1)	&3		8	(4+2)		&4				53 57.ms		55.a 741 417 472 271 247 712
&1 +		5		3	(2+2+1)	2	(3+1+1)	&4		8	(4+2)		&3				53 57.ms		55.b 645 513 456 351 263 632
&3 −		5		2	(3+1+1)	3	(2+2+1)	&3		9	(3+3)		&4				54 57.ms		56.a 721 427 172 274 147 714
&3 −		5		3	(2+2+1)	2	(3+1+1)	&4		9	(3+3)		&2				54 57.ms		56.b 546 315 465 531 236 362
&2 ∓		7		11	(3+2+2)	11	(3+2+2)	&6		8	(4+2)		&3				70 74.ab		72 657 753 567 735 376 673
&2 ∓		7		11	(3+2+2)	11	(3+2+2)	&6		14	(7)		&1				71 74.ab		73 375 576 763 637 736 367 573 Conjugacy classes: Inner circle 14b Outer circle 14a 675 357 756 765 537
&3 −		5 6		3  5	(2+2+1) (3+2+1)	3  5	(2+2+1) (3+2+1)	&5		14	(7)		&1				BL.56 BM.56 BR.56		B 372 276 543 615 731 467 463 Conjugacy classes: Inner circle 14a Outer circle 14b 631 256 352 715 547
&1 +		5 6		3  5	(2+2+1) (3+2+1)	3  5	(2+2+1) (3+2+1)	&5		9	(3+3)		&3				BL.56 TL.56		CL 732 267 643 613 371 476 453 651 253 652 517 745
&1 +		5 6		3  5	(2+2+1) (3+2+1)	3  5	(2+2+1) (3+2+1)	&5		8	(4+2)		&2				BM.56 TM.56		CM 726 327 526 325 175 574 165 534 136 364 137 764
&1 +		5 6		3  5	(2+2+1) (3+2+1)	3  5	(2+2+1) (3+2+1)	&5		14	(7)		&1				BR.56 TR.56		CR 672 273 345 516 751 457 436 Conjugacy classes: Inner circle 14b Outer circle 14a 361 265 532 713 647
&3 −		5 6		3  5	(2+2+1) (3+2+1)	3  5	(2+2+1) (3+2+1)	&5		9	(3+3)		&2				TL.56 TM.56 TR.56		T.a 762 237 346 316 571 475 435 561 235 562 317 746
&3 −		5 6		3  5	(2+2+1) (3+2+1)	3  5	(2+2+1) (3+2+1)	&5		9	(3+3)		&3				TL.56 TM.56 TR.56		T.b 627 723 523 625 157 754 163 634 156 354 173 674

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 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
&1 3				30 34		32 124 421
&1 3				31 34		33 142 241 412 214
&2 4				43.ab 40		41.a 143 341 542 245 416 614 512 215
&2 4				43.ab 40		41.b 342 243 152 251 612 216 452 254
&2 4				40 44		42.a 146 641 413 314
&2 4				40 44		42.b 162 261 432 234
&2 4				43.ab 47		45 164 461 126 621 431 134 423 324
&2 4				44 47		46 521 125 524 425
&3 5₂₃				51.ab 52.ab		50 174 471 326 623
&3 5₂₃				51.ab 55.ab		53 712 217 351 153 742 247 654 456
&3 5₂₃				52.ab 56.ab		54 724 427 135 531 721 127 465 564
&3 5₂₃				55.ab 56.ab		57.m 362 263 172 271 632 236 472 274
&3 5₂₃				55.ab 56.ab		57.s 714 417 315 513 741 147 645 546
&6 7				72		70 673 376
&6 7				73		71 736 637 375 573 763 367 675 576
&6 7				72 73		74.a 753 357 756 657
&6 7				72 73		74.b 735 537 765 567
&4 5₃₃				B CL		BL.5 352 253 652 256
&5 6				B CL		BL.6 715 517 745 547
&4 5₃₃				B CM		BM.5 631 136 364 463
&5 6				B CM		BM.6 731 137 764 467
&4 5₃₃				B CR		BR.5 615 516 345 543
&5 6				B CR		BR.6 372 273 672 276
&4 5₃₃				CL T.ab		TL.5 346 643 156 651 613 316 453 354
&5 6				CL T.ab		TL.6 732 237 371 173 762 267 674 476
&4 5₃₃				CM T.ab		TM.5 165 561 526 625 435 534 523 325
&5 6				CM T.ab		TM.6 726 627 175 571 723 327 475 574
&4 5₃₃				CR T.ab		TR.5 562 265 163 361 532 235 436 634
&5 6				CR T.ab		TR.6 713 317 751 157 746 647 754 457

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 14_a and 14_b. 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 14_a and six from 14_b.

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 14_a and 14_b 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 14_a and 14_b:

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.