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{{DISPLAYTITLE:Fischer group Fi<sub>22</sub>}}   
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In the area of modern algebra known as [[Group theory|group theory]], the '''Fischer group''' ''Fi<sub>22</sub>'' is a sporadic simple group of [[Order (group theory)|order]]
:&nbsp;&nbsp;&nbsp;2<sup>17</sup>{{·}}3<sup>9</sup>{{·}}5<sup>2</sup>{{·}}7{{·}}11{{·}}13
: = 64561751654400
: ≈ 6{{e|13}}.

==History==
''Fi<sub>22</sub>'' is one of the 26 sporadic groups and is the smallest of the three Fischer groups.  It was introduced by {{harvs|txt |first=Bernd |last=Fischer |year1=1971 |year2=1976}} while investigating [[3-transposition group|3-transposition group]]s.

The [[Outer automorphism group|outer automorphism group]] has order 2, and the [[Schur multiplier]] has order 6.

== Representations ==

The Fischer group Fi<sub>22</sub> has a rank 3 action  on  a graph of 3510 vertices corresponding to its 3-transpositions, with point stabilizer the double cover of the group PSU<sub>6</sub>(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism.

Fi<sub>22</sub> has an irreducible  real representation of dimension 78. Reducing an integral form of this mod 3 gives a representation of Fi<sub>22</sub> over the field with 3 elements, whose quotient by the 1-dimensional space of fixed vectors is a 77-dimensional irreducible representation.

The perfect triple cover of Fi<sub>22</sub> has an irreducible representation of dimension 27 over the field with 4 elements. This arises from the fact that Fi<sub>22</sub> is a subgroup of <sup>2</sup>E<sub>6</sub>(2<sup>2</sup>).
All the ordinary and modular character tables of Fi<sub>22</sub> have been computed. {{harvtxt|Hiss|White|1994}} found the 5-modular character table, and {{harvtxt|Noeske|2007}} found the 2- and 3-modular character tables.

The automorphism group of Fi<sub>22</sub> centralizes an element of order 3 in the [[Baby monster group|baby monster group]].

==Generalized Monstrous Moonshine==

Conway and Norton suggested in their 1979 paper that [[Monstrous moonshine|monstrous moonshine]] is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For ''Fi''<sub>22</sub>, the McKay-Thompson series is <math>T_{6A}(\tau)</math> where one can set a(0) = 10 ({{OEIS2C|id=A007254}}),

:<math>\begin{align}j_{6A}(\tau)
&=T_{6A}(\tau)+10\\
&=\left(\left(\tfrac{\eta(\tau)\,\eta(3\tau)}{\eta(2\tau)\,\eta(6\tau)}\right)^{3}+2^3 \left(\tfrac{\eta(2\tau)\,\eta(6\tau)}{\eta(\tau)\,\eta(3\tau)}\right)^{3}\right)^2\\
&=\left(\left(\tfrac{\eta(\tau)\,\eta(2\tau)}{\eta(3\tau)\,\eta(6\tau)}\right)^{2}+3^2 \left(\tfrac{\eta(3\tau)\,\eta(6\tau)}{\eta(\tau)\,\eta(2\tau)}\right)^{2}\right)^2-4\\
&=\frac{1}{q} + 10 + 79q + 352q^2 +1431q^3+4160q^4+13015q^5+\dots
\end{align}</math>

and ''η''(''τ'') is the [[Dedekind eta function]].

== Maximal subgroups ==
{{harvtxt|Wilson|1984}} found the 12 conjugacy classes of maximal subgroups of ''Fi<sub>22</sub>'' as follows:

* 2·U<sub>6</sub>(2)
* O<sub>7</sub>(3) (Two classes, fused by an outer automorphism)
* O{{su|b=8|p=+}}(2):S<sub>3</sub>
* 2<SUP>10</SUP>:M<sub>22</sub>
* 2<SUP>6</SUP>:S<sub>6</sub>(2)
* (2 × 2<SUP>1+8</SUP>):(U<sub>4</sub>(2):2)
* U<sub>4</sub>(3):2 × S<sub>3</sub>
* <SUP>2</SUP>F<sub>4</sub>(2)'  (This is the [[Tits group]])
* 2<SUP>5+8</SUP>:(S<sub>3</sub> × A<sub>6</sub>)
* 3<SUP>1+6</SUP>:2<SUP>3+4</SUP>:3<SUP>2</SUP>:2
* S<sub>10</sub> (Two classes, fused by an outer automorphism)
* M<sub>12</sub>

== References ==

*{{Citation | last1=Aschbacher | first1=Michael |  title=3-transposition groups | publisher={{wipe|Cambridge University Press}} | series=Cambridge Tracts in Mathematics | isbn=978-0-521-57196-8 | year=1997 | volume=124 |url=http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511759413 | mr=1423599 | doi=10.1017/CBO9780511759413}} contains a complete proof of Fischer's theorem.
*{{Citation | last1=Conway | first1=John Horton |  editor1-last=Shult | editor1-first=and Ernest E. | editor2-last=Hale | editor2-first=Mark P. | editor3-last=Gagen | editor3-first=Terrence | title=Finite groups '72 (Proceedings of the Gainesville Conference on Finite Groups, University of Florida, Gainesville, Fla., March 23–24, 1972.) | publisher=North-Holland | location=Amsterdam | series=North-Holland Mathematics Studies | mr=0372016  | year=1973 | volume=7 | chapter=A construction for the smallest Fischer group F<sub>22</sub> | pages=27–35}}
*{{Citation | last1=Fischer | first1=Bernd | title=Finite groups generated by 3-transpositions. I | doi=10.1007/BF01404633 | year=1971 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=13 | pages=232–246 | mr=0294487 | issue=3}} This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
*{{Citation | last1=Fischer | first1=Bernd | title=Finite Groups Generated by 3-transpositions | url=https://books.google.com/books?id=PjezNwAACAAJ | publisher=Mathematics Institute, University of Warwick | series=Preprint | year=1976}}
*{{Citation | last2=White | first2=Donald L. | last1=Hiss | first1=Gerhard | title=The 5-modular characters of the covering group of the sporadic simple Fischer group Fi<sub>22</sub> and its automorphism group | doi=10.1080/00927879408825043 | mr=1278807  | year=1994 | journal=Communications in Algebra | issn=0092-7872 | volume=22 | issue=9 | pages=3591–3611}}
*{{Citation | last1=Noeske | first1=Felix | title=The 2- and 3-modular characters of the sporadic simple Fischer group Fi<sub>22</sub> and its cover | doi=10.1016/j.jalgebra.2006.06.020 | mr=2303203  | year=2007 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=309 | issue=2 | pages=723–743| doi-access=free }}
*{{Citation | last1=Wilson | first1=Robert A. | title=On maximal subgroups of the Fischer group Fi<sub>22</sub> | doi=10.1017/S0305004100061491 | mr=735364  | year=1984 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=95 | issue=2 | pages=197–222}}
* {{Citation | last1=Wilson | first1=Robert A. | title=The finite simple groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | year=2009 | zbl=1203.20012 | volume=251}}
*Wilson, R. A.  [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/F22/ ATLAS of Finite Group Representations.]

== External links ==
* [http://mathworld.wolfram.com/FischerGroups.html MathWorld: Fischer Groups]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/lookup?target=Fi22&SUBMIT=Go Atlas of Finite Group Representations: Fi22]

{{DEFAULTSORT:Fischer group Fi22}}
[[Category:Sporadic groups]]

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{{Sourceattribution|Fischer group Fi22}}