Fischer group Fi23

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In mathematics, the Fischer group Fi23 or M(23) or F23 of order

218 · 313 · 52 · 7 · 11 · 13 · 17 · 23 (= 4089470473293004800) is one 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 1, and the Schur multiplier has order 1.

Representations[edit]

The Fischer group Fi23 has a rank 3 action on a graph of 31671 vertices corresponding to 3-transpositions, with point stabilizer the double cover of the Fischer group Fi22. It has a second rank-3 action on 137632 points

The smallest faithful complex representation has dimension 782. The group has an irreducible representation of dimension 253 over the field with 3 elements.

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 Fi23, the relevant McKay-Thompson series is T_{3A}(\tau) where one can set the constant term a(0) = 42 (OEISA030197),

\begin{align}j_{3A}(\tau)
&=T_{3A}(\tau)+42\\
&=\Big(\big(\tfrac{\eta(\tau)}{\eta(3\tau)}\big)^{6}+3^3 \big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{6}\Big)^2\\
&=\frac{1}{q} + 42 + 783q + 8672q^2 +65367q^3+371520q^4+\dots
\end{align}

and η(τ) is the Dedekind eta function. Let r_n = 1, 782, 3588, 5083, 25806, 30888, 60996,...(OEISA003914) be the degrees of irreducible representations of Fi23. Then,

\begin{align}
1 &= r_1\\
783 &= r_1+r_2\\
8672 &= r_1+r_3+r_4\\
65367 &= r_1+r_2+r_3+r_7\\
371520 &= r_1 + 2 r_2 + r_3 + r_4 + r_5 + r_7 + r_{10}\\
\end{align}

and so on. Like the j-function, the function j3A(τ) can also assume an integer value for appropriate arguments. For example,

j_{3A}\Big(\tfrac{3+\sqrt{-267}}{6}\Big) = -300^3

Maximal subgroups[edit]

Kleidman, Parker & Wilson (1989) found the conjugacy classes of maximal subgroups of Fi23 as follows:

2.Fi22

O8+(3):S3

22.U6(2).2

S8(2)

O7(3) × S3

211.M23

31+8.21+6.31+2.2S4

[310].(L3(3) × 2)

S12

(22 × 21+8).(3 × U4(2)).2

26+8:(A7 × S3)

S6(2) × S4

S4(4):4

L2(23)

References[edit]