In mathematics, the Hessian group is a finite group of order 216, introduced by 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 3<sup>2</sup>, and the quotient by this subgroup is isomorphic to the group SL<sub>2</sub>(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, <sub>3</sub>[3]<sub>3</sub>[3]<sub>3</sub> or of order 648, and the product of this with a group of order 2 is another complex reflection group, <sub>3</sub>[3]<sub>3</sub>[4]<sub>2</sub> or of order 1296.