Chalcogenides are becoming increasingly important in the development of new solid-state materials for technological applications. Chalcogenide clusters with well-defined size and composition represent the lower limit of semiconducting nanoparticles and serve to span the size gap between quantum dot structures and molecular species in solution.[1] Moreover, these large clusters can be used as building blocks to construct supramolecular assemblies with unique properties. Among chalcogenide clusters with various geometrical features, [2][3][4] tetrahedral clusters are of particular significance because they can act as pseudotetrahedral building blocks for the construction of zeolite-like open architectures. [5][6][7][8][9] Unfortunately, even though a number of superlattices built from supertetrahedral clusters have been reported, [10][11][12][13] relatively little progress has been made with other types of tetrahedral clusters.Here we report a three-dimensional open-framework material (denoted ICF-26) built from an unusual tetrahedral cluster. ICF-26 was prepared from the Ca-Li-In-S quaternary system in a procedure mimicking the preparation of natural zeolites by using alkali and alkaline earth metal cations as structure-directing agents.[14] Organic species are not needed as either surface-stabilizing ligands or extraframework structure-directing agents, in contrast to other recent work that relies heavily on the use of organic compounds. [15][16][17][18][19][20][21] The highly mobile extraframework cations results in the ionic conductivity of ICF-26 being higher than that of any previously known crystalline lithium compound at room temperature. [22,23] ICF-26 is built from a tetrahedral cluster denoted P2 (Figure 1). The P2 cluster (that is, Li 4 In 22 S 44 18À ) is the second member of a mathematical series of pentasupertetrahedral clusters Pn, thus termed because they can be conceptually constructed by coupling four supertetrahedral clusters onto the faces of an antisupertetrahedral cluster of the same order (Figure 1). Supertetrahedral clusters are regular tetrahedronshaped fragments of the cubic ZnS type lattice and are denoted as Tn, where n is the number of metal layers.[10] An antisupertetrahedral cluster is defined here as having the same geometrical features as a supertetrahedral cluster with the positions of cations and anions being exchanged.Thus, the P1 cluster consists of four T1 clusters (MX 4 ) at the corners and one anti-T1 cluster (XM 4 ) at the core, resulting in the composition ( A pentasupertetrahedral cluster is considerably larger than a supertetrahedral cluster of the same order, and hence it is difficult to prepare open-framework materials with pentasupertetrahedral clusters larger than P1. Even though supertetrahedral clusters as large as T5 are known, [12] the P2 cluster reported herein represents the largest known cluster of the Pn series.While all metal cations of each P2 cluster adopt tetrahedral coordination, sulfur atoms can be two-, three-, or fourcoordinate. There are a total of f...