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1-wreath with parameters 3-connected abelian group array automorphism Baer subplanes bipartite graph canonical form Catalan Cayley graph cc(G cells chains of length clique covering Combinatorial complete mapping components construction contains contradiction Corollary corresponding coset cubic graph cycle defined deleted denote Desarguesian planes direct sum edge Edited elements equation equivalent example exists finite fully embedded GF(q group G group Hadamard matrix Hadamard matrix Hence Hughes type hyperplanes idempotent incident induction integers intersection isomorphic J.A. Thas K-system kronecker product LCBGs Lemma Math Mathematics neofield nodes number of points obtained orthogonal design ovals partial difference set partial geometries partially transitive permutation planar plane of order poset problem Proceedings projective plane Proof Rogers algebra Room square row and column Schroder sequence subgroup subsets subsquare Suppose symbols Theorem Theory translation plane type 6,m u-tree University of Adelaide vector vertex vertices