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{{DISPLAYTITLE:<sup>3</sup>D<sub>4</sub>}}In mathematics, the Steinberg triality [[group (mathematics)|groups]] of type '''<sup>3</sup>D<sub>4</sub>''' form a family of [[Steinberg group (Lie theory)|Steinberg]] or [[twisted Chevalley group]]s. They are [[quasi-split group|quasi-split]] forms of D<sub>4</sub>, depending on a cubic [[Galois extension]] of [[field (mathematics)|fields]] ''K'' ⊂ ''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 group|orthogonal]] or [[spin group]] in dimension&nbsp;8.

Over [[finite field]]s these groups form one of the 18 infinite families of [[finite simple group]]s, and were introduced  by  {{harvtxt|Steinberg|1959}}. They were independently discovered by [[Jacques Tits]] in {{harvtxt|Tits|1958}} and {{harvtxt|Tits|1959}}.

==Construction==

The [[simply connected]] split algebraic group of type D<sub>4</sub> has a triality automorphism σ of order 3 coming from an order 3 [[automorphism]] of its Dynkin diagram. If ''L'' is a field with an automorphism τ of order 3, then this induced an order 3 automorphism τ 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 στ.  It has three 8-dimensional representations over the field ''L'', [[permutation|permuted]] by the outer automorphism τ of order&nbsp;3.

==Over finite fields==

The group <sup>3</sup>D<sub>4</sub>(''q''<sup>3</sup>) has order
''q''<sup>12</sup>
(''q''<sup>8</sup>&nbsp;+&nbsp;''q''<sup>4</sup>&nbsp;+&nbsp;1)
(''q''<sup>6</sup>&nbsp;−&nbsp;1)
(''q''<sup>2</sup>&nbsp;−&nbsp;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>&nbsp;−&nbsp;2''q''<sup>4</sup>&nbsp;+&nbsp;1)
(''q''<sup>6</sup>&nbsp;−&nbsp;1)
(''q''<sup>2</sup>&nbsp;−&nbsp;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>&nbsp;−&nbsp;1)
(''q''<sup>6</sup>&nbsp;−&nbsp;1)
(''q''<sup>2</sup>&nbsp;−&nbsp;1).

The group <sup>3</sup>D<sub>4</sub>(''q''<sup>3</sup>) is always [[simple group|simple]].  The [[Schur multiplier]] is always trivial.  The [[outer automorphism group]] is [[cyclic group|cyclic]] of order ''f'' where ''q''<sup>3</sup> = ''p<sup>f</sup>'' and ''p'' is [[prime number|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.

==<sup>3</sup>D<sub>4</sub>(2<sup>3</sup>)==

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>⋅3<sup>4</sup>⋅7<sup>2</sup>⋅13 and outer automorphism group of order&nbsp;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 {{harvtxt|Gross|Elkies|1996}}. 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
: 2<SUP>1+8</SUP>:L<SUB>2</SUB>(8) fixing a point of the rank 4 permutation representation on 819 points.
: [2<SUP>11</SUP>]:(7 × S<SUB>3</SUB>)
: U<SUB>3</SUB>(3):2
: S<SUB>3</SUB> × L<SUB>2</SUB>(8)
: (7 × L<SUB>2</SUB>(7)):2
: 3<SUP>1+2</SUP>.2S<SUB>4</SUB>
: 7<SUP>2</SUP>:2A<SUB>4</SUB>
: 3<SUP>2</SUP>:2A<SUB>4</SUB>
: 13:4

==See also==

*[[List of finite simple groups]]
*[[2E6 (mathematics)|<sup>2</sup>E<sub>6</sub>]]

==References==

*{{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Simple groups of Lie type | origyear=1972 | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-50683-6 | mr=0407163 | year=1989}}
*{{Citation | last1=Elkies | first1=Noam D. | last2=Gross | first2=Benedict H. | title=The exceptional cone and the Leech lattice | doi=10.1155/S1073792896000426 | mr=1411589  | year=1996 | journal=International Mathematics Research Notices | issn=1073-7928 | issue=14 | pages=665–698}}
*{{Citation | last1=Steinberg | first1=Robert | title=Variations on a theme of Chevalley | url=http://projecteuclid.org/euclid.pjm/1103039126 | mr=0109191 | year=1959 | journal=[[Pacific Journal of Mathematics]] | issn=0030-8730 | volume=9 | pages=875–891 | doi=10.2140/pjm.1959.9.875}}
*{{Citation|last1=Steinberg |first1=Robert |title=Lectures on Chevalley groups |url=http://www.math.ucla.edu/~rst/ |publisher=Yale University, New Haven, Conn. |mr=0466335 |year=1968 |deadurl=yes |archiveurl=https://web.archive.org/web/20120910032654/http://www.math.ucla.edu/~rst/ |archivedate=2012-09-10 }}
*{{Citation | last1=Tits | first1=Jacques | title=Les "formes réelles" des groupes de type E<sub>6</sub> | url=http://www.numdam.org/item?id=SB_1956-1958__4__351_0 | publisher=Secrétariat math'ematique | location=Paris | series=Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162 | mr=0106247 | year=1958 | volume=15}}
*{{Citation | last1=Tits | first1=Jacques | title=Sur la trialité et certains groupes qui s'en déduisent | journal = Inst. Hautes Etudes Sci. Publ. Math. | url = https://link.springer.com/article/10.1007%2FBF02684706 | year=1959 | volume=2 | pages = 13–60}}

==External links==
*[http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TD42/ ³D₄(2³) at the atlas of finite groups]
*[http://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TD43/ ³D₄(3³) at the atlas of finite groups]

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[[Category:Finite groups]]
[[Category:Lie groups]]