The numbers ${\displaystyle {\mathit {0}}\dots {\mathit {23}}}$ represent the permutations of 4 elements in reverse colexicographic order (compare Symmetric group S₄).

In the position ${\displaystyle (a,b)}$ this table shows the result of ${\displaystyle a^{b}}$ (a^b) in SymPy, which is the same as ${\displaystyle b^{-1}\cdot a\cdot b}$ (~b*a*b).

There is also: left action last ⋅ first

This is: right action first ⋅ last

In terms of left action, usual as function composition, this would be ${\displaystyle b\cdot a\cdot b^{-1}}$.

The leftmost column serves as a vertical index. To the left the matrices of the permutations are shown.

The second horizontal index (semi-transparent) shows the inverse permutations of those in the main index
With ${\displaystyle c}$ taken from the second index, position ${\displaystyle (a,c)}$ shows ${\displaystyle c\cdot a\cdot c^{-1}}$ (or ${\displaystyle c^{-1}\cdot a\cdot c}$ in terms of left action).

Example: ${\displaystyle {\mathit {6^{15}}}~~=~~{\mathit {15}}^{-1}\cdot {\mathit {6}}\cdot {\mathit {15}}~~=~~{\mathit {20}}\cdot {\mathit {6}}\cdot {\mathit {20}}^{-1}~~=~~{\mathit {20}}\cdot {\mathit {6}}\cdot {\mathit {15}}~~=~~{\mathit {2}}}$              (${\displaystyle {\mathit {15}}^{-1}={\mathit {20}}}$)

>>> from sympy.combinatorics import Permutation
>>> Permutation.print_cyclic = False
>>> a = Permutation([0, 1, 3, 2])
>>> b = Permutation([3, 0, 2, 1])
>>> a^b
Permutation[0, 2, 1, 3]
>>> ~b*a*b
Permutation[0, 2, 1, 3]

Calculation of the matrix with Python: http://pastebin.com/BGkvfdgH (as exp_table and conjugacy_table1)

The entries in the same row are in the same conjugacy class. Every entry appears once in each column, as seen in the file below.

The positions of permutations with inversion sets symmetric to each other have positions in the table that are symmetric to each other.
This symmetry is seen in the cycle graph on the right.
E.g. ${\displaystyle {\mathit {17}}}$ and ${\displaystyle {\mathit {22}}}$ are symmetric to each other.
${\displaystyle {\mathit {0}}}$ (trivial), ${\displaystyle {\mathit {2}}}$, ${\displaystyle {\mathit {21}}}$ (green), ${\displaystyle {\mathit {10}}}$, ${\displaystyle {\mathit {13}}}$ (orange) and ${\displaystyle {\mathit {7}}}$, ${\displaystyle {\mathit {16}}}$, ${\displaystyle {\mathit {23}}}$ (bold) are mirror-symmetric.