D_n is planar exactly for n in {1, p, p^2, p^3, pq, p^2 q}, and its characteristic polynomial divides that of D_{n p q}, with a square divisibility when n has a simple prime factor.
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On divisibility relation graphs
D_n is planar exactly for n in {1, p, p^2, p^3, pq, p^2 q}, and its characteristic polynomial divides that of D_{n p q}, with a square divisibility when n has a simple prime factor.