We consider socially structured transferable utility games. For every coalition the relative strength of a player is measured by a power function. The socially stable core consists of the socially and economically stable payoff vectors. It is non-empty if the game itself is socially stable. The socially stable core consists of a finite number of faces of the core. Generically, it consists of a finite number of payoff vectors. Convex tu-games have a non-empty socially stable core, irrespective of the underlying social structure. When the game is permutationally convex, the socially stable core is shown to be non-empty if the power vectors are permutationally consistent and is shown to contain a unique element if the power vectors are permutationally compatible. We demonstrate the usefulness of the concept of the socially stable core by applying it to structured hierarchy games, sequencing games and the distribution of water.