A transitive triple, \((a, b, c)\), is defined to be the set \(\{(a, b), (b, c), (a, c)\}\) of ordered pairs. A directed triple system of order \(v\), \(DTS(v)\), is a pair \((D, \beta)\), where \(D\) is a set of \(v\) points and \(\beta\) is a collection of transitive triples of pairwise distinct points of \(D\) such that any ordered pair of distinct points of \(D\) is contained in precisely one transitive triple of \(\beta\). An antiautomorphism of a directed triple system, \((D, \beta)\), is a permutation of \(D\) which maps \(\beta\) to \(\beta^{-1}\), where \(\beta^{-1} = \{(c, b, a) | (a, b, c) \in \beta\}\). In this paper, we give necessary and sufficient conditions for the existence of a directed triple system of order \(v\) admitting an antiautomorphism consisting of two cycles of lengths \(M\) and \(2M\), and one fixed point.
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