Coherent configurations (example and explanation) A coherent configuration is a
pair $(X, S)$ consisting of a finite set $X $ of size $v$  and a set $S$ of binary relations on $X$ such that

•$ S$ is a partition of $X × X$;

• the diagonal relation $ ∆X$  is the union of some relations in $S$;

• for each $R  \in  S$ it holds that $R^{T}  \in S$;

• there exist integers $p^{R}_{ ST}$ such that
|$\{z ∈ X|(x, z)  \in  S$  and  $(z, y) \in  T \}| = p^{R}_{ ST} \}$
whenever $(x, y) \in  R$, for each $R, S, T ∈ S$.

I am new to this subject, so it is hard for me to understand. I request to explain above four axioms using an example.

I have tried some notes, most of them have not included any example.
The canonical examples arise from a group of permutations acting on $X$. Define two ordered pairs $(a,x)$ and $(b,y)$ to be related if there is an element of $G$ that maps $(a,x)$ to $(b,y)$, i.e., some element $g$ in $G$ maps $a$ to $b$ and $x$ to $y$. The set of ordered pairs related to a given pair is an orbit of $G$ acting on $X\times X$, and these orbits form a partition of $X\times X$. Thus the first axiom holds.

For the second axiom, note that all pairs in the orbit of a pair $(a,a)$ have the form $(b,b)$, and so the diagonal of $X\times X$ has a partition as required.

The orbit of $(a,b)$ is the ``transpose'' of the orbit of $(b,a)$. 

Finally, if $x$ is $S$-related to $a$ and $T$-related to $b$ and $g\in G$, then $x^g$ is also $S$-related $a^g$ and $T$-related to $b^g$. So, in what I hope is an obvious notation,
\[ p_{S,T}(a,b) = p_{S,T}(a^g,b^g)\]
for all $g$ in $G$. From this it follows that the intersection numbers are well-defined.
We also know that if each relation R is represented by a matrix A whose rows and columns are indexed by the elements of X with (x,y) entry 1 if (x,y) belongs to R , otherwise 0.
Then the three conditions become:
1. The sum of the matrices is the all-1 matrix.
2.There is a subset of the matrices which sums to the identity matrix.
3.The set of matrices is closed under transposition.

But what will the fourth condition become ? Please explain in detail.