The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Abstract Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner systems, little is known about automorphism groups. In particular, there no congruence class representing orders of KTS number automorphisms at least close to points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ? 39 (mod 72), or 4^e48 + 3$$ 4 e 48 + 3 $$4^e96$$ 96 ) $$e \ge 0$$ ? 0 , exists on v points having $$v-3$$ - automorphisms. This only one consequences an investigation KTSs group G acting sharply transitively all but three Our methods constructive yield which many cases inherit some thus increasing total symmetries. To obtain these results it was necessary introduce new types difference families (the doubly disjoint ones) matrices splittable we believe interesting themselves.