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As Dima Pasechnik says, every irreducible module for a finite group $G$ occurs as a composition factor of the regular representation of $G$, that is the permutation module coming from the regular permutation action of $G$ on itself. In this question Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? , it is pointed out that the only finite groups which have complex irreducible modules not occurring in any permutation module on any non-trivial subgroup of $G$ are the so-called Frobenius complements.

Later note: In fact, over an algebraically closed field $F$ of characteristic $p$, every irreducible $FG$-module occurs as a composition factor of the permutation module on the cosets of a Sylow $p$-subgroup of $G.$ (In fact, each occurs both in the socle and the head of that permutation module).

As Dima Pasechnik says, every irreducible module for a finite group $G$ occurs as a composition factor of the regular representation of $G$, that is the permutation module coming from the regular permutation action of $G$ on itself. In this question Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? , it is pointed out that the only finite groups which have complex irreducible modules not occurring in any permutation module on any non-trivial subgroup of $G$ are the so-called Frobenius complements.

Later note: In fact, over an algebraically closed field $F$ of characteristic $p$, every irreducible $FG$-module occurs as a composition factor of the permutation module on the cosets of a Sylow $p$-subgroup of $G.$ (In fact, each occurs both in the socle and the head of that permutation module).

As Dima Pasechnik says, every irreducible module for a finite group $G$ occurs as a composition factor of the regular representation of $G$, that is the permutation module coming from the regular permutation action of $G$ on itself. In this question Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? , it is pointed out that the only finite groups which have complex irreducible modules not occurring in any permutation module on any non-trivial subgroup of $G$ are the so-called Frobenius complements.

Later note: In fact, over an algebraically closed field $F$ of characteristic $p$, every irreducible $FG$-module occurs as a composition factor of the permutation module on the cosets of a Sylow $p$-subgroup of $G$ In fact, each occurs both in the socle and the head of that permutation module).

As Dima Pasechnik says, every irreducible module for a finite group $G$ occurs as a composition factor of the regular representation of $G$, that is the permutation module coming from the regular permutation action of $G$ on itself. In this question Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? , it is pointed out that the only finite groups which have complex irreducible modules not occurring in any permutation module on any non-trivial subgroup of $G$ are the so-called Frobenius complements.

Later note: In fact, over an algebraically closed field $F$ of characteristic $p$, every irreducible $FG$-module occurs as a composition factor of the permutation module on the cosets of a Sylow $p$-subgroup of $G.$ (In fact, each occurs both in the socle and the head of that permutation module).

As Dima Pasechnik says, every irreducible module for a finite group $G$ occurs as a composition factor of the regular representation of $G$, that is the permutation module coming from the regular permutation action of $G$ on itself. In this question Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? , it is pointed out that the only finite groups which have complex irreducible modules not occurring in any permutation module on any non-trivial subgroup of $G$ are the so-called Frobenius complements.

As Dima Pasechnik says, every irreducible module for a finite group $G$ occurs as a composition factor of the regular representation of $G$, that is the permutation module coming from the regular permutation action of $G$ on itself. In this question Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? , it is pointed out that the only finite groups which have complex irreducible modules not occurring in any permutation module on any non-trivial subgroup of $G$ are the so-called Frobenius complements.

Later note: In fact, over an algebraically closed field $F$ of characteristic $p$, every irreducible $FG$-module occurs as a composition factor of the permutation module on the cosets of a Sylow $p$-subgroup of $G$ In fact, each occurs both in the socle and the head of that permutation module).