If $G$ is a transitive subgroup of $S_n$, then we draw $n$ dots to represent the numbers from $1$ to $n$. For each generator of the group, we draw an arrow from the $i$th dot to the $j$th dot if the permutation sends $i$ to $j$. There are no arrows drawn for fixed points, and a transposition is represented by a single double-headed arrow. One color is assigned to each generator for its arrows.
For groups where the diagram has not yet been created, we display a picture of Évariste Galois.
Implementation
Diagrams were created by John Jones using the package graphviz.