Valentiner group

Representations

• The alternating group A₆ acts on the complex projective plane, and showed that the group acts on the 6 conics of Gerbaldi's theorem. This gives a homomorphism to PGL₃, and the lift of this to the triple cover GL₃ is the Valentiner group. This embedding can be defined over the field generated by the 15th roots of unity.
• The product of the Valentiner group with a group of order 2 is a 3-dimensional complex reflection group of order 2160 generated by 45 complex reflections of order 2. The invariants form a polynomial algebra with generators of degrees 6, 12, and 30.
• The Valentiner group has complex irreducible faithful group representations of dimension 3, 3, 3, 3, 6, 6, 9, 9, 15, 15.
• The Valentiner group can be represented as the monomial symmetries of the hexacode, the 3-dimensional subspace of F spanned by,, and, where the elements of the finite field F₄ are 0, 1, ω,.
• The group PGL₃ acts on the 2-dimensional projective plane over F₄ and acts transitively on its hyperovals. The subgroup fixing a hyperoval is a copy of the alternating group A₆. The lift of this to the triple cover GL₃ of PGL₃ is the Valentiner group.
• described the representations of the Valentiner group as a Galois group, and gave an order 3 differential equation with the Valentiner group as its differential Galois group.