3-bit Walsh permutation

Walsh permutation

Binary Walsh matrix

Fano plane with nimber labels

There are  A002884(3) = 4 * 6 * 7 = 168 invertible binary 3×3 matrices.

They form the general linear group GL(3,2). (As all non-zero determinants are 1 in the binary field, it is also the special linear group.)

It is isomorphic to the projective special linear group PSL(2,7), the symmetry group of the Fano plane.

Each of these maps corresponds to a permutation of seven elements, which can be seen as a collineation of the Fano plane.

Conjugacy classes

The group has six conjugacy classes. They almost correspond to the cycle type,
but there are two different conjugacy classes with 7-cycles. (For the distinction between them, see here.)

description	neutral	2+2	2+4	3+3	7a	7b
size	1	21	42	56	24	24
examples	124	421 bit-reversal	136 Gray code	241	765	357
		623	247	517	512	243

The 3×3 matrices in the same conjugacy class are similar.

example of matrix similarity
$P^{-1}$ $A$ $P$ $=$ $B$ matrix multiplication 127 461 127 $=$ 753 147 461 172 $=$ 753 751 461 436 $=$ 753 731 461 463 $=$ 753

Cycle shapes

In a symmetric representation like the Fano plane, there are 33 different cycle shapes (or 24 if the direction is ignored).
A complete list can be found here. They are denoted by city names, followed by a qualifier of the direction, where needed.

triangle Berlin
217 742 471

hexagon Shanghai
236 362 315 546 531 465

triangles Rio + and −
Inverse 3×3 matrices differ in their determinant. Those in the inner triangle have 1, and those in the outer have −1. 567 735 657 753 673 376

hexagons Lima 5 and 6
Inverse 3×3 matrices differ in their number of ones. Those in the inner hexagon have 5, and those in the outer have 6. 137 764 136 364 327 726 325 526 534 165 574 175

$A$ and $B$ have the same cycle shape, if there is a $P$ from the symmetric subgroup, so that $B=P^{-1}AP$. (The 3×3 matrix of $P$ is a permutation matrix.)

This is a refinement of matrix similarity, where $P$ is allowed to be any element of the group. (Cycle shapes are a refinement of conjugacy classes.)

Powers and cycle graph

The cycle graph of this group has 28 triangles, 21 squares and 8 heptagons.

Each of the following rows is an example of a cycle. (Each one is closed by the neutral element, which is not shown.)
It shows consecutive powers of the first element from left to right. (Also of the last element from right to left.)
Elements in symmetric positions are inverse to each other.

Triangles

Each of the 28 triangles contains two inverse permutations of cycle type 3+3.

Squares

This shows why there are two permutations of cycle type 2+4 for each one with 2+2.

Heptagons

Alexandria, Kinshasa, Cairo
713 576 452 341 637 265 Alexandria a Kinshasa a Cairo b Cairo a Kinshasa b Alexandria b

Kinshasa, Cairo, Alexandria
763 251 647 532 416 375 Kinshasa b Cairo b Alexandria a Alexandria b Cairo a Kinshasa a