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We study extremal black hole solutions of the S3 model (obtained by settingS=T=U in the STU model) using group theoretical methods. Upon dimensionalreduction over time, the S3 model exhibits the pseudo-Riemannian cosetstructure G/K with G=G2(2) and K=SO(2,2). We study nilpotent K-orbits of G2(2)corresponding to non-rotating single-center extremal solutions. We find sixsuch distinct K-orbits. Three of these orbits are supersymmetric, one isnon-supersymmetric, and two are unphysical. We write general solutions anddiscuss examples in all four physical orbits. We show that all solutions insupersymmetric orbits when uplifted to five-dimensional minimal supergravityhave single-center Gibbons-Hawking space as their four-dimensional Euclideanhyper-K\"ahler base space. We construct hitherto unknown extremal(supersymmetric as well as non-supersymmetric) pressureless black strings ofminimal five-dimensional supergravity and briefly discuss their relation toblack rings.

Extremal solutions of the S3 model and nilpotent orbits of G2(2)