 Diagonal action on a centerless group

For nontrivial centerless $T$, this action of $T\times T$ is faithful and transitive. Its identity stabilizer is the diagonal subgroup. Subgroups above that diagonal are $\{(a,b):a^{-1}b\in N\}$ with $N⊴T$: multiply by a diagonal element to isolate the second coordinate, then use diagonal conjugation. Thus the action is primitive exactly when $T$ is nonabelian simple.