In mathematics, the Steinberg triality groups of type <sup>3</sup>D<sub>4</sub> form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D<sub>4</sub>, depending on a cubic Galois extension of fields K â L, and using the triality automorphism of the Dynkin diagram D<sub>4</sub>. Unfortunately the notation for the group is not standardized, as some authors write it as <sup>3</sup>D<sub>4</sub>(K) (thinking of <sup>3</sup>D<sub>4</sub> as an algebraic group taking values in K) and some as <sup>3</sup>D<sub>4</sub>(L) (thinking of the group as a subgroup of D<sub>4</sub>(L) fixed by an outer automorphism of order 3). The group <sup>3</sup>D<sub>4</sub> is very similar to an orthogonal or spin group in dimension 8.
Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced by . They were independently discovered by Jacques Tits in and .
The simply connected split algebraic group of type D<sub>4</sub> has a triality automorphism ÃÂ of order 3 coming from an order 3 automorphism of its Dynkin diagram. If L is a field with an automorphism ÃÂ of order 3, then this induced an order 3 automorphism ÃÂ of the group D<sub>4</sub>(L). The group <sup>3</sup>D<sub>4</sub>(L) is the subgroup of D<sub>4</sub>(L) of points fixed by ÃÂÃÂ. It has three 8-dimensional representations over the field L, permuted by the outer automorphism ÃÂ of order 3.
The group <sup>3</sup>D<sub>4</sub>(q<sup>3</sup>) has order q<sup>12</sup> (q<sup>8</sup> + q<sup>4</sup> + 1) (q<sup>6</sup> â 1) (q<sup>2</sup> â 1). For comparison, the split spin group D<sub>4</sub>(q) in dimension 8 has order q<sup>12</sup> (q<sup>8</sup> â 2q<sup>4</sup> + 1) (q<sup>6</sup> â 1) (q<sup>2</sup> â 1) and the quasisplit spin group <sup>2</sup>D<sub>4</sub>(q<sup>2</sup>) in dimension 8 has order q<sup>12</sup> (q<sup>8</sup> â 1) (q<sup>6</sup> â 1) (q<sup>2</sup> â 1).
The group <sup>3</sup>D<sub>4</sub>(q<sup>3</sup>) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclic of order f where q<sup>3</sup> = p<sup>f</sup> and p is prime.
This group is also sometimes called <sup>3</sup>D<sub>4</sub>(q), D<sub>4</sub><sup>2</sup>(q<sup>3</sup>), or a twisted Chevalley group.
The smallest member of this family of groups has several exceptional properties not shared by other members of the family. It has order 211341312 = 2<sup>12</sup>â 3<sup>4</sup>â 7<sup>2</sup>â 13 and outer automorphism group of order 3.
The automorphism group of <sup>3</sup>D<sub>4</sub>(2<sup>3</sup>) is a maximal subgroup of the Thompson sporadic group, and is also a subgroup of the compact Lie group of type F<sub>4</sub> of dimension 52. In particular it acts on the 26-dimensional representation of F<sub>4</sub>. In this representation it fixes a 26-dimensional lattice that is the unique 26-dimensional even lattice of determinant 3 with no norm 2 vectors, studied by . The dual of this lattice has 819 pairs of vectors of norm 8/3, on which <sup>3</sup>D<sub>4</sub>(2<sup>3</sup>) acts as a rank 4 permutation group.
The group <sup>3</sup>D<sub>4</sub>(2<sup>3</sup>) has 9 classes of maximal subgroups, of structure