Hesse configuration

A Hesse configuration is a set $P$ of nine non-collinear points in the projective plane over a field $K$ such that any line through two points of $P$ contains exactly three points of $P$. Then there are 12 such lines through $P$. A Hesse configuration exists if and only if the field $K$ contains a primitive third root of unity. For such $K$ the projective automorphism group $\mathrm{PGL}(3,K)$ acts transitively on all possible Hesse configurations.

The configuration $P$ with its intersection structure of 12 lines is isomorphic to the affine space $A={𝔽}^2$ where $𝔽$ is a field with three elements.

The group $\mathrm{\Gamma }\subset \mathrm{PGL}(3,K)$ of all symmetries that map $P$ onto itself has order 216 and it is isomorphic to the group of affine transformations of $A$ that have determinant 1. The stabilizer in $\mathrm{\Gamma }$ of any of the 12 lines through $P$ is a cyclic subgroup of order three and $\mathrm{\Gamma }$ is generated by these subgroups.

The symmetry group $\mathrm{\Gamma }$ is isomorphic to $G(K)/Z(K)$ where $G(K)\subset \mathrm{GL}(3,K)$ is a group of order 648 generated by reflections of order three and $Z(K)$ is its cyclic center of order three. The reflection group $G(ℂ)$ is called the Hesse group which appears as $G_{25}$ in the classification of finite complex reflection groups by Shephard and Todd.

If $K$ is algebraically closed and the characteristic of $K$ is not 2 or 3 then the nine inflection points of an elliptic curve $E$ over $K$ form a Hesse configuration.