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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.^{[1]} It has a normal subgroup that is an elementary abelian group of order 3^{2}, and the quotient by this subgroup is isomorphic to the group SL_{2}(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.

- 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

**^**Hessian group on GroupNames