has a group of order 1296 which may be generated by the following set of eight substitutions: {ABC), {AB), (DEF), (DE), (GHI), (GH), (ADG)(BEH) (CFI), (AD)(BE)(CF). 2 Kagno's proof of this fact (Theorem S) 1 is straightforward, but somewhat lengthy, and it seems of interest to note that this theorem follows at once from a more general and almost self-evident theorem on the groups of repeated graphs, if we apply to Pappus' graph the following lemma, also due to Kagno: 1 "If G' is the complement of G, then G and G f have the same group." 3 Indeed the complement IF of Pappus' graph contains the 9 arcs AB, AC, BC, DE, DF, EF, GH, GI, HI; hence IF is not connected, but consists of three triangles (or complete 3-points) ABC, DEF, GHI; that is, II' is a threefold repeated triangle. To such a repeated graph we can apply the following theorem, which is of interest in itself apart from the use made of it here. THEOREM. If G is a connected graph of n vertices, having no simple loops, with a group & of order h, and if T is the graph consisting of m copies Gi, G 2 , • • • , G m of the same graph G, then the group of Y is Pôlya's "Gruppenkranz" ©m[^p], that is, the group of order m\h m and degree mn, whose substitutions may be described briefly as follows'. 4 ' Let