# Symmetric group S4

The symmetric group $S_{4}$ is the group of all permutations of 4 elements.
It has $4!=4\cdot 3\cdot 2\cdot 1=24$ elements and is not abelian.
Even permutations are white:
* the identity
* eight 3-cycles
* three double-transpositions (in bold typeface)
Odd permutations are colored:
* six transpositions (green)
* six 4-cycles (orange)
The small table on the left shows the permuted elements, and inversion vectors (which are reflected factorial numbers) below them.
Another column shows the inversion sets, ordered like . (When a dot with the numbers i,j is marked red, than the elements on places i,j are out of their natural order.)
The digit sums of the inversion vectors (or factorial numbers) and the cardinalities of the inversion sets are equal. They form the sequence  A034968. A permutation and its corresponding digit sum have the same parity.
The big table on the right is the Cayley table of S$_4$. It could also be given as the matrix multiplication table of the shown permutation matrices. (Compare multiplication table for S$_3$)
| Permutations of 4 elements | Cayley table of S$_4$ See also: A closer look at the Cayley table |

## Subgroups

There are 30 subgroups of S$_4$, including the group itself and the 10 small subgroups.
Every group has as many small subgroups as neutral elements on the main diagonal: The trivial group and two-element groups Z$_2$. These small subgroups are not counted in the following list.

### Order 8

| Dihedral group of order 8 Subgroups: | Dihedral group of order 8 Subgroups: | Dihedral group of order 8 Subgroups: |
Cycle graphs :
| Dihedral group of order 8 Subgroups: | Dihedral group of order 8 Subgroups: | Dihedral group of order 8 Subgroups: |

### Order 6

| Symmetric group S$_3$ Subgroup: | Symmetric group S$_3$ Subgroup: | Symmetric group S$_3$ Subgroup: | Symmetric group S$_3$ Subgroup: |
Cycle graphs :
| Symmetric group S$_3$ Subgroup: | Symmetric group S$_3$ Subgroup: | Symmetric group S$_3$ Subgroup: | Symmetric group S$_3$ Subgroup: |

### Order 4

| Klein four-group | Klein four-group | Klein four-group | Klein four-group |
| Cyclic group Z$_4$ | Cyclic group Z$_4$ | Cyclic group Z$_4$ |
Cycle graphs :
| Klein four-group | Klein four-group | Klein four-group | Klein four-group |
| Cyclic group Z$_4$ | Cyclic group Z$_4$ | Cyclic group Z$_4$ |

### Order 3

| Cyclic group Z$_3$ | Cyclic group Z$_3$ | Cyclic group Z$_3$ | Cyclic group Z$_3$ |
Cycle graphs :
| Cyclic group Z$_3$ | Cyclic group Z$_3$ | Cyclic group Z$_3$ | Cyclic group Z$_3$ |

## Lattice of subgroups

The subgroups of every group form a lattice:

## Weak order of permutations

The permutations of n elements form a lattice. A permutation may be defined by its set of inversions;
and the lattice by the subset relation $\subseteq$ between these sets. Or a permutation my be defined by its factorial number (or inversion vector); and the lattice by the bitwise less than or equal relation $\leq$ between them.

### Permutohedron

The Hasse diagram of the weak order of permutations is the permutohedron. For the symmetric group S$_4$ it's the truncated octahedron.
| Permutohedron of S$_4$ | Permutations represented by their sets of inversions |
| Schlegel diagram of the truncated octahedron | In the Schlegel diagram, the inversions show a remarkable symmetry. |
| Permutations represented by matrices | Permutations represented by permuted elements, and inversion vectors below them (compare convex version) |

### Join and meet

If one wants to have join and meet of any two permutations, one can find them in the permutohedron. The following table shows all relevant pairs of permutations. The arguments (row and column of the table) and their join and meet are shown by red vertices in the little permutohedra. The highest red vertex is always the join $\cup$ and the lowest red vertex is the meet $\cap$.
The following join table is derived from the table above. Besides the decimal enumeration, it shows also the inversion sets and factorial numbers. (The meet table is like this one, but reflected about the subdiagonal, and with all numbers replaced by their difference with 23.)
It's worth taking a look at the number of red squares in the 24 matrices above:
Number of entries in join and meet table :
| This permutohedron shows, how often an element appears in the join table - i.e. how many red squares are in the 24 matrices above. | And these are the numbers for the meet table. |
The table becomes more interesting, looking only at the inversion bits. This file shows the bits of every inversion in a single matrix:
One can see, that for the three inversions comparing consecutive places, the join operation is simply the union of the inversion sets. For the two inversions comparing places with one place between, there are 32 darker red fields beyond the union. For the inversion comparing the first and the last place, there are 56 darker fields beyond the union. So the union of two permutations inversion sets is always a subset to the inversion set of the permutations' join.

## A closer look at the Cayley table

Every entry appears exactly one time in every row and column of the Cayley table. So the positions of the entries form 24 permutation matrices:
Rows and columns of the Cayley table match permutations of 24 elements. Below they are also represented as permutation matrices:

## Generators and Cayley graphs
### Truncated octahedron
#### Generators 1, 9 or (12), (1234)
#### Generators 1, 2, 6 or (12), (23), (34)

* (compare convex version)
Cayley graph and permutohedron :
| This graph is related to the permutohedron: Self inverse permutations (symmetric matrices) are on the same positions, all other permutations are replaced by their inverses (transposed matrices). | Permutohedron for comparison |

### Nauru graph - Generators 1, 5, 21 or (12), (13), (14)
#### Torus embedding

* Sequences with the first digit omitted Triples linked by an edge differ in only one digit. The edge color tells by which: 1st, 2nd, 3rd

#### Adjacency matrix
## Bit permutations

When the permutations p$_n$ of 4 elements are applied on the reverse binary digits of the integers 0...15, they generate permutations P$_n$ of 16 elements, which also form the symmetric group S$_4$. Walsh permutation; bit permutation

## Gray code order (Steinhaus–Johnson–Trotter algorithm)

| The algorithm defines a Hamiltonian path in a Cayley graph of the symmetric group. The inverse permutations define a path in the permutohedron: | |
| Cayley graph | Permutohedron |
| Permutations form a Gray code. The swapped elements are always adjacent. | Permutations, inversion vectors and inversion sets form a Gray code. |
| Permutations with green or orange background are odd. Numbers in boldface type are double-transpositions. The smaller numbers below the permutations are the inversion vectors. Red marks indicate swapped elements. ||