# 3-bit Walsh permutation

Fano plane with nimber labels

There are (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.)

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

## 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.

hexagon Shanghai

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

This is a refinement of matrix similarity, where ${\displaystyle P}$ 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.

Toronto, Rome
 Toronto Rome Toronto

### Heptagons

Cairo, Alexandria, Kinshasa
 Cairo b Alexandria b Kinshasa a Kinshasa b Alexandria a Cairo a