Hessian group

In mathematics, the Hessian group is a finite group of order 216, introduced by Jordan (1877) who named it for Otto Hesse. It may be represented as the group of affine transformations with determinant 1 of the affine plane over the finite field of 3 elements. It has a normal subgroup that is an elementary abelian group of order 3², and the quotient by this subgroup is isomorphic to the group SL₂(3) of order 24. It also acts on the Hesse pencil of elliptic curves, and forms the automorphism group of the Hesse configuration of the 9 inflection points of these curves and the 12 lines through triples of these points.

The triple cover of this group is a complex reflection group, ₃[3]₃[3]₃ or of order 648, and the product of this with a group of order 2 is another complex reflection group, ₃[3]₃[4]₂ or of order 1296.

References

• Artebani, Michela; Dolgachev, Igor (2009), "The Hesse pencil of plane cubic curves", L'Enseignement Mathématique, 2e Série, 55 (3): 235–273, arXiv:math/0611590, doi:10.4171/lem/55-3-3, ISSN 0013-8584, MR 2583779
• Coxeter, Harold Scott MacDonald (1956), "The collineation groups of the finite affine and projective planes with four lines through each point", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 20: 165–177, doi:10.1007/BF03374555, ISSN 0025-5858, MR 0081289
• Grove, Charles Clayton (1906), The syzygetic pencil of cubics with a new geometrical development of its Hesse Group, Baltimore, Md.
• Jordan, Camille (1877), "Mémoire sur les équations différentielles linéaires à intégrale algébrique.", Journal für die reine und angewandte Mathematik (in French), 84: 89–215, doi:10.1515/crll.1878.84.89, ISSN 0075-4102