= Solution
The \b[$z$ axis is a $C_2$ axis] through the centers of opposite cube faces: a half turn permutes atoms within each species. A \b[body diagonal through sites 1 and 8 is a $C_3$ axis]: a third turn fixes those two opposite vertices and cycles the other three vertices of each tetrahedral sublattice. These are <symmetry operations> of the <tetrahedral symmetry> group $T_d$; a quarter turn about $z$ would interchange the two chemical species and is not a symmetry of the specified cluster.
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