The line phases in the title are obtained in high yields after fusion of the elements in stoichiometric proportions in Ta containers followed by slow cooling or annealing. The structures were established by single-crystal X-ray means [Na(4)A(6)Tl(13), A = K, Rb, Cs : Im(3) over bar, Z = 2, a 11.5075(7) Angstrom, 11.6648(5) Angstrom, 11.8332(4) Angstrom, R(w)(F) = 2.9, 4.9, 4.2%, respectively; Na3K8Tl13 : R(3) over barm$, Z = 3, a 11.0890(6) Angstrom, c = 23.154(2) Angstrom, R(w) = 4.9%]. The structure of cubic Na4K6Tl13 is practically identical with that reported for NaK9Tl13, a composition that cannot be achieved in this structure. The two structure types contain isolated, centered icosahedra of Tl-13(10-) (T-h) and Tl-13(11-)(D-3d) in bcc and ccp arrangements, respectively. All cations bridge between cluster faces or vertices, and their mixed sizes appear critical to the formation of the two structures containing these clusters; which do not occur in the binary systems. The phases exhibit Curie-Weiss (1.7, 1.9 mu(B)) (cubic Rb, Cs) and temperature-independent (chi(M) similar to 1.9 x 10(-3) emu/mol) (rhombohedral K) paramagnetism and poor metal-like resistivities (rho(293) = 310, 625, 69 mu Omega cm), respectively. The novel one-electron-deficient Tl-13(10-) is a result of cation packing limitations, while Tl-13(11-) fulfills (Wade’s rule) predictions for a centered icosahedron. Centering of the clusters by thallium contributes markedly to the bonding in both. Theoretical calculations at the extended Huckel and relativistic Huckel (REX) levels for the clusters and at the LMTO (self-consistent band) level for Na4Rb6Tl13 are all consistent with the experimental finding. Mixing of 6p(1/2) and 6p(3/2) levels in the clusters reduces relativistic separations substantially (to similar to 0.2 eV) relative to that in the free Tl atom. Electron localization within the clusters appears dominant.