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. 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 . 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.