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For a transitive finite $G$-set $X$ with at least two points, the augmentation subrepresentation of a permutation representation is irreducible if and only if the action is two-transitive. Indeed, writing the permutation character as $\pi =1+{\sum }_i{m}_i{\chi }_i$ givesThe augmentation subrepresentation is irreducible exactly when this norm is two. By the character norm of a permutation representation, that means that the diagonal and the off-diagonal are the only two orbits on $X\times X$, which is exactly two-transitivity.

The vectoris fixed by every element of $G$, so $\mathbb{C}{1}_X\cong {\mathbb{C}}_G$ is a trivial representation. The coefficient-sum subspaceis also $G$-invariant, and every vector has a unique decomposition into a constant vector and an element of $V$. HenceHere $V$ is the augmentation subrepresentation of a permutation representation.