Fischer group Fi22

For general background and history of the Fischer sporadic groups, see Fischer group.

In the area of modern algebra known as group theory, the Fischer group Fi₂₂ is a sporadic simple group of order

   2¹⁷ · 3⁹ · 5² · 7 · 11 · 13

= 64561751654400

≈ 6×10¹³.

History

Fi₂₂ is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups.

The outer automorphism group has order 2, and the Schur multiplier has order 6.

Representations

The Fischer group Fi₂₂ 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 PSU₆(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism.

Fi₂₂ has an irreducible real representation of dimension 78. Reducing an integral form of this mod 3 gives a representation of Fi₂₂ 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 Fi₂₂ has an irreducible representation of dimension 27 over the field with 4 elements. This arises from the fact that Fi₂₂ is a subgroup of ²E₆(2²). All the ordinary and modular character tables of Fi₂₂ have been computed. Hiss & White (1994) found the 5-modular character table,and Noeske (2007) found the 2- and 3-modular character tables.

The automorphism group of Fi₂₂ centralizes an element of order 3 in the baby monster.

Generalized Monstrous Moonshine

Conway and Norton suggested in their 1979 paper that 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 Fi₂₂, the McKay-Thompson series is $T_{6A}(\tau)$ where one can set a(0) = 10 ( A007254),

$\begin{align}j_{6A}(\tau) &=T_{6A}(\tau)+10\\ &=\Big(\big(\tfrac{\eta(\tau)\,\eta(3\tau)}{\eta(2\tau)\,\eta(6\tau)}\big)^{3}+2^3 \big(\tfrac{\eta(2\tau)\,\eta(6\tau)}{\eta(\tau)\,\eta(3\tau)}\big)^{3}\Big)^2\\ &=\Big(\big(\tfrac{\eta(\tau)\,\eta(2\tau)}{\eta(3\tau)\,\eta(6\tau)}\big)^{2}+3^2 \big(\tfrac{\eta(3\tau)\,\eta(6\tau)}{\eta(\tau)\,\eta(2\tau)}\big)^{2}\Big)^2-4\\ &=\frac{1}{q} + 10 + 79q + 352q^2 +1431q^3+4160q^4+13015q^5+\dots \end{align}$

and η(τ) is the Dedekind eta function.

Maximal subgroups

Wilson (1984) found the 12 conjugacy classes of maximal subgroups of Fi₂₂ as follows:

• 2·U₆(2)
• O₇(3) (Two classes, fused by an outer automorphism)
• O⁺
  ₈(2):S₃
• 2¹⁰:M₂₂
• 2⁶:S₆(2)
• (2 × 2¹⁺⁸):(U₄(2):2)
• U₄(3):2 × S₃
• ²F₄(2)'
• 2⁵⁺⁸:(S₃ × A₆)
• 3¹⁺⁶:2³⁺⁴:3²:2
• S₁₀ (Two classes, fused by an outer automorphism)
• M₁₂

References

• contains a complete proof of Fischer's theorem.
• 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).
• Wilson, R. A. ATLAS of Finite Group Representations.

External links

• MathWorld: Fischer Groups
• Atlas of Finite Group Representations: Fi22