Fischer group Fi22

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In mathematics, the Fischer group Fi22 or M(22) or F22, of order

217 · 39 · 52 · 7 · 11 · 13 (= 64561751654400)

is the smallest of the three Fischer groups, sporadic simple groups 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[edit]

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[edit]

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 (OEISA007254),

\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. Let r_n = 1, 78, 429, 1001, 1430, 3003, 3080, 10725, 13650,...(OEISA003913) be the degrees of irreducible representations of Fi22. Then,

\begin{align}
1 &= r_1\\
79 &= r_1+r_2\\
352&= r_1-r_2+r_3\\
1431&= r_1 + r_3 + r_4\\
4160&= 2r_2 + r_4 + r_6\\
13015&= 2r_1 + 2r_3 + r_5 + r_8\\
\end{align}

and so on. For a specific value,

j_{6A}\Big(\tfrac{3+\sqrt{-177}}{6}\Big) = -1060^2

which implies,

e^{\pi/3\sqrt{177}} = 1060^2 + 9.999929\dots

Maximal subgroups[edit]

Wilson (1984) found the 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[edit]