3-bit Walsh permutation/matrix columns

There are 25 transforms that look similar to the neutral position.
This is the case when the view from one or two axes remains the same or almost the same.
Almost the same means, that the original square is sheared into a (simple) parallelogram.

Cluster of the neighbor graph, on the right the matrix sums and vertex types
124
125 (as inverse)
136 (binary)
125 and 136 look similar to the neutral position.
125 looks the same from x and almost the same from y (here shown from −y).
136 looks almost the same from x.

These are the 25 permutations in the middle cluster of the positive component of the neighbor graph.
(There are only 18 transforms that do not look like a square or simple parallelogram from any side, namely those who's matrices have seven 1s.)


The big table below shows all 168 transforms in 28 rows and 6 columns.

The shown transforms are those with binary matrices. For the transforms with matrices that have some negative entries see here.

For the 25 rows with sums < 7 the position in the table corresponds with that in the neighbor graph: Left, middle or right cluster in the positive or negative component.


conjugacy class neut. 2+2 2+4 7a
cycle shape Paris Rome Florence Buenos Aires 5 Buenos Aires 6 Santiago +
sum 3 4 5a 5b 6 7
quantity 1 6 3 3 6 6 3
12 12


Permutations in the same row have the same complement pattern, and each complement pattern corresponds to four rows.

The big table can be sorted by some properties of the permutations in the main column:

  • cc conjugacy class
  • cs cycle shape: abbreviated (city) name of the cycle shape; below the sum (3...7), which corresponds to the position in the cluster (which is why 5a and 5b are distinguished)
  • cp complement pattern: above the weight (1...3), below the value (1...7)
  • t triple of numbers used in the vectors in the row (number in ascending order)

Let be a permutation from the symmetric subgroup in the top row, and a permutation in the main column.
Then the permutation in the row of and the column of is .     E.g. 314 (second row, negative left column) is 134214.

cc cs cp t main positive negative
&0 neut. &0 neut. 3 37 124


241

124

412

214

421

142
2+2 &1 P 4 26 134


341

134

413

314

431

143
2+2 &1 P 4 26 125


251

125

512

215

521

152
2+2 &1 P 4 25 234


243

324

432

234

423

342
2+2 &1 P 4 25 126


261

126

612

216

621

162
2+2 &1 P 4 23 245


245

524

452

254

425

542
2+2 &1 P 4 23 146


641

164

416

614

461

146
2+2 &2 R 5a 11 247


247

724

472

274

427

742
2+2 &2 R 5a 12 147


741

174

417

714

471

147
2+2 &2 R 5a 14 127


271

127

712

217

721

172
2+2 &3 F 5a 37 135


351

135

513

315

531

153
2+2 &3 F 5a 37 236


263

326

632

236

623

362
2+2 &3 F 5a 37 456


645

564

456

654

465

546
2+4 &4 BA 5 5b 12 156


651

165

516

615

561

156
2+4 &4 BA 5 5b 14 136


361

136

613

316

631

163
2+4 &4 BA 5 5b 14 235


253

325

532

235

523

352
2+4 &4 BA 5 5b 11 256


265

526

652

256

625

562
2+4 &4 BA 5 5b 12 345


345

534

453

354

435

543
2+4 &4 BA 5 5b 11 346


643

364

436

634

463

346
2+4 &5 BA 6 6 23 157


751

175

517

715

571

157
2+4 &5 BA 6 6 25 137


371

137

713

317

731

173
2+4 &5 BA 6 6 23 267


267

726

672

276

627

762
2+4 &5 BA 6 6 26 237


273

327

732

237

723

372
2+4 &5 BA 6 6 25 467


647

764

476

674

467

746
2+4 &5 BA 6 6 26 457


745

574

457

754

475

547
7a &6 S +7 11 357


537

753

375

573

357

735
7a &6 S +7 12 367


736

673

367

763

376

637
7a &6 S +7 14 567


576

657

765

567

756

675