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There are 71 Steiner triple systems of order 27 whose automorphism groups are point-transitive, and there are 248 transitive Kirkman triple systems of order 27. Computational methods used to find these designs are outlined. The designs and some of their properties are presented. 1. STEINER TRIPLE SYSTEMS AND THEIR GROUPS A Steiner triple system of order v, briefly STS(v), is a pair (V, q) where V is a set of v elements and q is a set of 3-element subsets of V, with the property that every 2-subset of V appears in exactly one subset of q . Sets in q' are triples. An automorphism of an STS(v) is a permutation on V that maps each triple in q' to a triple of W, and the automorphism group is the group of all automorphisms of the STS. The STS is transitive if the automorphism group acts transitively on V; it is cyclic when the group contains Z, as a subgroup, and abelian if there is an abelian subgroup of the automorphism group that acts transitively on V. Steiner triple systems with large automorphism groups, and in particular transitive STS, are studied in large part because they yield examples of Steiner triple systems with interesting "regularity." There is an extensive literature on cyclic STS [2], but for the closely related case of transitive STS, especially nonabelian STS, little is known. Transitive STS of order 21 have been constructively enumerated [5]; there are seven cyclic designs and three other transitive ones. Subsequently, Tonchev [8] constructively enumerated the transitive STS(25); there are three over Z5 X Z5, in addition to the twelve over Z25 known since the 1930s [1]. For order 27, there are five possible automorphism groups to be considered. Three are the abelian groups Z3 X Z3 X Z3, Z9 X Z3, and Z27. In addition, there are two nonabelian groups [3]. The first of these has all group elements of order 3, while the second has Z9 as a subgroup. Every transitive STS(27) has (at least) one of these contained in its automorphism group. Cyclic STS(27) 's have been generated previously [1], as have the "l-rotational" STS(27) 's [7]. In this paper, we exhibit all transitive STS(27)'s. There are eight cyclic ones [1]; we find that there are 71 transitive STS(27) 's in total. In addition to presenting these designs, we give an extensive computational analysis of them. Received June 21, 1990; revised January 15, 1991. 1991 Mathematics Subject Classification. Primary 05B07, 5 lEl0. ? 1992 American Mathematical Society 0025-5718/92 $1.00 + $.25 per page

Transitive Steiner and Kirkman triple systems of order 27