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 field of 3 elements.[1] 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