Abstract
The block-transitive point-imprimitive 2-(729,8,1) designs are classified. They all have full automorphism group of order 729.13 which is an extension of a groupN of order 729, acting regularly on points, by a group of order 13. There are, up to isomorphism, 27 designs withN elementary abelian, 13 designs withN=Z 39 and 427 designs withN the relatively free 3-generator, exponent 3, nilpotency class 2 group, a total of 467 designs. This classification completes the classification of block-transitive, point-imprimitive 2-(ν, k, 1) designs satisfying\(\upsilon = \left( {\left( {_2^k } \right) - 1} \right)^2\), which is the Delandtsheer-Doyen upper bound for the numberν of points of such designs. The only examples of block-transitive, point-imprimitive 2-(ν, k, 1) designs with\(\upsilon = \left( {\left( {_2^k } \right) - 1} \right)^2\) are the 2-(729, 8, 1) designs constructed in this paper.
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The first three authors acknowledge the support of an Australian European Awards Program scholarship, a Deutsche Akademische Austauschdienst scholarship, and an Australian Research Council Research Fellowship, respectively
The authors wish to thank Brendan McKay for his independent verification of the non-isomorphism of the classes of designs found, and of their automorphism groups, using different, “nauty” techniques [6].
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Nickel, W., Niemeyer, A.C., O'Keefe, C.M. et al. The block-transitive, point-imprimitive 2-(729, 8, 1) designs. AAECC 3, 47–61 (1992). https://doi.org/10.1007/BF01189023
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DOI: https://doi.org/10.1007/BF01189023