# 3D4 > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/3D4 > Markdown URL: https://mediated.wiki/source/3D4.md > Source: https://en.wikipedia.org/wiki/3D4 > Source revision: 1346220879 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Family of mathematical groups}} {{DISPLAYTITLE:³D₄}}In mathematics, the Steinberg triality [groups](/source/group_(mathematics)) of type '''³D₄''' form a family of [Steinberg](/source/Steinberg_group_(Lie_theory)) or [twisted Chevalley group](/source/twisted_Chevalley_group)s. They are [quasi-split](/source/quasi-split_group) forms of D₄, depending on a cubic [Galois extension](/source/Galois_extension) of [fields](/source/field_(mathematics)) ''K'' ⊂ ''L'', and using the [triality](/source/triality) automorphism of the [Dynkin diagram](/source/Dynkin_diagram) D₄. Unfortunately the notation for the group is not standardized, as some authors write it as ³D₄(''K'') (thinking of ³D₄ as an [algebraic group](/source/algebraic_group) taking values in ''K'') and some as ³D₄(''L'') (thinking of the group as a subgroup of D₄(''L'') fixed by an [outer automorphism](/source/outer_automorphism) of order 3). The group ³D₄ is very similar to an [orthogonal](/source/orthogonal_group) or [spin group](/source/spin_group) in dimension 8. Over [finite field](/source/finite_field)s these groups form one of the 18 infinite families of [finite simple group](/source/finite_simple_group)s, and were introduced by {{harvtxt|Steinberg|1959}}. They were independently discovered by [Jacques Tits](/source/Jacques_Tits) in {{harvtxt|Tits|1958}} and {{harvtxt|Tits|1959}}. ==Construction== The [simply connected](/source/simply_connected) split algebraic group of type D₄ has a triality automorphism σ of order 3 coming from an order 3 [automorphism](/source/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 D₄(''L''). The group ³D₄(''L'') is the subgroup of D₄(''L'') of points fixed by στ. It has three 8-dimensional representations over the field ''L'', [permuted](/source/permutation) by the outer automorphism τ of order 3. ==Over finite fields== The group ³D₄(''q''³) has order ''q''¹² (''q''⁸ + ''q''⁴ + 1) (''q''⁶ − 1) (''q''² − 1). For comparison, the split spin group D₄(''q'') in dimension 8 has order ''q''¹² (''q''⁸ − 2''q''⁴ + 1) (''q''⁶ − 1) (''q''² − 1) and the quasisplit spin group ²D₄(''q''²) in dimension 8 has order ''q''¹² (''q''⁸ − 1) (''q''⁶ − 1) (''q''² − 1). The group ³D₄(''q''³) is always [simple](/source/simple_group). The [Schur multiplier](/source/Schur_multiplier) is always trivial. The [outer automorphism group](/source/outer_automorphism_group) is [cyclic](/source/cyclic_group) of order ''f'' where ''q''³ = ''p^f'' and ''p'' is [prime](/source/prime_number). This group is also sometimes called ³''D''₄(''q''), ''D''₄²(''q''³), or a twisted Chevalley group. ==³D₄(2³)== The smallest member of this family of groups has several exceptional properties not shared by other members of the family. It has order 211341312 = 2¹²⋅3⁴⋅7²⋅13 and outer automorphism group of order 3. The automorphism group of ³D₄(2³) is a maximal subgroup of the [Thompson sporadic group](/source/Thompson_sporadic_group), and is also a subgroup of the compact [Lie group](/source/Lie_group) of type F₄ of dimension 52. In particular it acts on the 26-dimensional representation of F₄. 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|Elkies|Gross|1996}}. The dual of this lattice has 819 pairs of vectors of norm 8/3, on which ³D₄(2³) acts as a rank 4 [permutation group](/source/permutation_group). The group ³D₄(2³) has 9 classes of maximal subgroups, of structure : 2¹⁺⁸:L₂(8) fixing a point of the rank 4 permutation representation on 819 points. : [2¹¹]:(7 × S₃) : U₃(3):2 : S₃ × L₂(8) : (7 × L₂(7)):2 : 3¹⁺².2S₄ : 7²:2A₄ : 3²:2A₄ : 13:4 ==See also== *[List of finite simple groups](/source/List_of_finite_simple_groups) *[²E₆](/source/2E6_(mathematics)) ==References== *{{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Simple groups of Lie type | orig-year=1972 | publisher=[John Wiley & Sons](/source/John_Wiley_%26_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 | volume=1996 | issn=1073-7928 | issue=14 | pages=665–698 | doi-access=free }} *{{Citation | last1=Steinberg | first1=Robert | title=Variations on a theme of Chevalley | url=https://projecteuclid.org/euclid.pjm/1103039126 | mr=0109191 | year=1959 | journal=[Pacific Journal of Mathematics](/source/Pacific_Journal_of_Mathematics) | issn=0030-8730 | volume=9 | issue=3 | pages=875–891 | doi=10.2140/pjm.1959.9.875| doi-access=free }} *{{Citation|last1=Steinberg |first1=Robert |title=Lectures on Chevalley groups |url=https://www.math.ucla.edu/~rst/ |publisher=Yale University, New Haven, Conn. |mr=0466335 |year=1968 |url-status=dead |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 E₆ | url=https://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 Études Sci. Publ. Math. | year=1959 | volume=2 | pages = 13–60| doi=10.1007/BF02684706 | s2cid=120426125 | url=https://www.numdam.org/item/PMIHES_1959__2__13_0/ }} ==External links== *[https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TD42/ ³D₄(2³) at the atlas of finite groups] *[https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TD43/ ³D₄(3³) at the atlas of finite groups] Category:Finite groups Category:Lie groups --- Adapted from the Wikipedia article [3D4](https://en.wikipedia.org/wiki/3D4) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/3D4?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.