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| {{Group theory sidebar}}
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| In the [[mathematical]] field of [[group theory]], the '''Rudvalis group''' ''Ru'', found by {{Harvard citations |last=Rudvalis |first=Arunas |author-link=Arunas Rudvalis |year=1973 |year2=1984 |txt=yes}} and constructed by {{Harvard citations |last1=Conway |last2=Wales |year=1973 |txt=yes}}, is a [[Sporadic group|sporadic]] [[simple group]] of [[order (group theory)|order]]
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| : 2<sup>14</sup>{{·}}3<sup>3</sup>{{·}}5<sup>3</sup>{{·}}7{{·}}13{{·}}29
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| : = 145926144000
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| : ≈ 10<sup>11</sup>.
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| ''Ru'' is one of the six sporadic simple groups known as "[[pariah group]]s" as they are not found within the [[Monster group]] {{Harvard citations |last=Griess |year=1983 |loc=p. 91}}.
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| == Properties ==
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| The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the [[Ree group]]
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| <sup>2</sup>''F''<sub>4</sub>(2), the automorphism group of the [[Tits group]]. This representation implies a [[strongly regular graph]] in which each vertex has 2304 neighbors and 1755 non-neighbors. Two adjacent vertices have 1328 common neighbors; two non-adjacent ones have 1208 {{Harvard citations|last=Griess|year=1998|loc=p. 125}}
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| Its [[Schur multiplier]] has order 2, and its [[outer automorphism group]] is trivial.
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| Its [[Double covering group|double cover]] acts on a 28-dimensional lattice over the [[Gaussian integer]]s. The lattice has 4×4060 minimal vectors; if minimal vectors are identified if one is 1, ''i'', –1, or –''i'' times another then the 4060 equivalence classes can be identified with the points of the rank 3 permutation representation. Reducing this lattice modulo the [[principal ideal]]
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| :<math>(1 + i)\ </math>
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| gives an action of the Rudvalis group on a 28-dimensional vector space over the field with 2 elements. Duncan (2006) used the 28-dimensional lattice to construct a [[vertex operator algebra]] acted on by the double cover.
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| {{harvtxt|Parrott|1976}} characterized the Rudvalis group by the centralizer of a central involution. {{harvtxt|Aschbacher|Smith|2004}} gave another characterization as part of their identification of the Rudvalis group as one of the [[quasithin group]]s.
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| == Maximal subgroups ==
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| {{harvtxt|Wilson|1984}} found the 15 classes of maximal subgroups of the Rudvalis group, as follows:
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| <SUP>2</SUP>F<SUB>4</SUB>(2) = <SUP>2</SUP>F<SUB>4</SUB>(2)'.2,
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| 2<SUP>6</SUP>.U<SUB>3</SUB>(3).2,
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| (2<SUP>2</SUP> × Sz(8)):3,
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| 2<SUP>3+8</SUP>:L<SUB>3</SUB>(2),
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| U<SUB>3</SUB>(5):2,
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| 2<SUP>1+4+6</SUP>.S<SUB>5</SUB>,
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| PSL<SUB>2</SUB>(25).2<SUP>2</SUP>,
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| A<SUB>8</SUB>,
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| PSL<SUB>2</SUB>(29),
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| 5<SUP>2</SUP>:4.S<SUB>5</SUB>,
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| 3.A<SUB>6</SUB>.2<SUP>2</SUP>,
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| 5<SUP>1+2</SUP>:[2<SUP>5</SUP>],
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| L<SUB>2</SUB>(13):2,
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| A<SUB>6</SUB>.2<SUP>2</SUP>,
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| 5:4 × A<SUB>5</SUB>.
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| ==References ==
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| *{{Citation | last1=Aschbacher | first1=Michael | author1-link=Michael Aschbacher | last2=Smith | first2=Stephen D. | title=The classification of quasithin groups. I Structure of Strongly Quasithin K-groups | url=http://www.ams.org/bookstore-getitem/item=SURV-111 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3410-7 | mr=2097623 | year=2004 | volume=111}}
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| * {{Citation
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| |last1 = Conway
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| |first1 = J.H.
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| |author1-link = John H. Conway
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| |last2 = Wales
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| |first2 = D.B.
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| |title = The construction of the Rudvalis simple group of order 145926144000
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| |journal = Journal of Algebra
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| |issue = 27
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| |year = 1973
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| |pages = 538–548
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| |doi = 10.1016/0021-8693(73)90063-X
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| |volume = 27
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| }}
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| * {{cite arXiv
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| |author = John F. Duncan
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| |title = Moonshine for Rudvalis's sporadic group
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| |year = 2008
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| |class = math.RT
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| |version = v1
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| |eprint = math/0609449
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| }}
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| * {{Citation
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| |last = Griess
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| |first = R.L.
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| |author-link = R. L. Griess
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| |title = The Friendly Giant
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| |journal = Inventiones Mathematicae
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| |issue = 69
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| |year = 1982
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| |pages = 1–102
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| |doi = 10.1007/BF01389186
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| |volume = 69
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| }}
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| * {{Citation
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| |last = Griess
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| |first = R.L.
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| |title = Twelve Sporadic Groups
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| |year = 1998
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| |publisher = Springer-Verlag
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| }}
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| *{{Citation | last1=Parrott | first1=David | title=A characterization of the Rudvalis simple group | doi=10.1112/plms/s3-32.1.25 | mr=0390043 | year=1976 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=32 | issue=1 | pages=25–51}}
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| * {{Citation
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| |last = Rudvalis
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| |first = A.
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| |author-link = Arunas Rudvalis
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| |title = A new simple group of order 2<sup>14</sup> 3<sup>3</sup> 5<sup>3</sup> 7 13 29
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| |journal = Notices of the American Mathematical Society
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| |issue = 20
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| |year = 1973
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| |pages = A–95
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| }}
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| *{{Citation | last1=Rudvalis | first1=Arunas | title=A rank 3 simple group of order 2¹⁴3³5³7.13.29. I | url=http://dx.doi.org/10.1016/0021-8693(84)90063-2 | doi=10.1016/0021-8693(84)90063-2 | id={{MR|727376}} | year=1984 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=86 | issue=1 | pages=181–218}}
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| *{{Citation | last1=Rudvalis | first1=Arunas | title=A rank 3 simple group G of order 2¹⁴3³5³7.13.29. II. Characters of G and Ĝ | url=http://dx.doi.org/10.1016/0021-8693(84)90064-4 | doi=10.1016/0021-8693(84)90064-4 | id={{MR|727377}} | year=1984 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=86 | issue=1 | pages=219–258}}
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| *{{Citation | last1=Wilson | first1=Robert A. | title=The geometry and maximal subgroups of the simple groups of A. Rudvalis and J. Tits | doi=10.1112/plms/s3-48.3.533 | mr=735227 | year=1984 | journal=Proceedings of the London Mathematical Society. Third Series | issn=0024-6115 | volume=48 | issue=3 | pages=533–563}}
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| ==External links==
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| * [http://mathworld.wolfram.com/RudvalisGroup.html MathWorld: Rudvalis Group]
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Ru/ Atlas of Finite Group Representations: Rudvalis group]
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| [[Category:Sporadic groups]]
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