  We study extremal black hole solutions of the S3 model (obtained by setting
S=T=U in the STU model) using group theoretical methods. Upon dimensional
reduction over time, the S3 model exhibits the pseudo-Riemannian coset
structure 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 six
such distinct K-orbits. Three of these orbits are supersymmetric, one is
non-supersymmetric, and two are unphysical. We write general solutions and
discuss examples in all four physical orbits. We show that all solutions in
supersymmetric orbits when uplifted to five-dimensional minimal supergravity
have single-center Gibbons-Hawking space as their four-dimensional Euclidean
hyper-K\"ahler base space. We construct hitherto unknown extremal
(supersymmetric as well as non-supersymmetric) pressureless black strings of
minimal five-dimensional supergravity and briefly discuss their relation to
black rings.
