Valentiner group

In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by in the form of an action of A₆ on the complex projective plane, and was studied further by .

All perfect alternating groups have perfect double covers. In most cases this is the universal central extension. The two exceptions are A₆ (whose perfect triple cover is the Valentiner group) and A₇, whose universal central extensions have centers of order 6.

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(C), and the lift of this to the triple cover SL₃(C) 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 (001111), (111100), and (0101ω), where the elements of the finite field F₄ are 0, 1, ω, .
• The group PGL₃(F₄) acts on the 2-dimensional projective plane over F₄ and acts transitively on its hyperovals (sets of 6 points such that no three are on a line). The subgroup fixing a hyperoval is a copy of the alternating group A₆. The lift of this to the triple cover GL₃(F₄) of PGL₃(F₄) 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.

References