Fischer group Fi23

In the area of modern algebra known as group theory, the Fischer group Fi23 is a sporadic simple group of order
 * 218·313·52·7·11·13·17·23
 * = 4089470473293004800
 * ≈ 4.

History
Fi23 is one of the 26 sporadic groups and is one of the three Fischer groups introduced by while investigating 3-transposition groups.

The Schur multiplier and the outer automorphism group are both trivial.

Representations
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

Fi23 is the centralizer of a transposition in the Fischer group Fi24. When realizing Fi24 as a subgroup of the Monster group, the full centralizer of a transposition is the double cover of the Baby monster group. As a result, Fi23 is a subgroup of the Baby monster, and is the normalizer of a certain S3 group in the Monster.

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
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 ,


 * $$\begin{align}j_{3A}(\tau)

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

and η(τ) is the Dedekind eta function.

Maximal subgroups
found the 14 conjugacy classes of maximal subgroups of Fi23 as follows:


 * 2.Fi22
 * O8+(3):S3
 * 22.U6(2).2
 * S8(2)
 * O7(3) × S<SUB>3</SUB>
 * 2<SUP>11</SUP>.M<SUB>23</SUB>
 * 3<SUP>1+8</SUP>.2<SUP>1+6</SUP>.3<SUP>1+2</SUP>.2S<SUB>4</SUB>
 * [3<SUP>10</SUP>].(L<SUB>3</SUB>(3) × 2)
 * S<SUB>12</SUB>
 * (2<SUP>2</SUP> × 2<SUP>1+8</SUP>).(3 × U<SUB>4</SUB>(2)).2
 * 2<SUP>6+8</SUP>:(A<SUB>7</SUB> × S<SUB>3</SUB>)
 * S<SUB>6</SUB>(2) × S<SUB>4</SUB>
 * S<SUB>4</SUB>(4):4
 * L<SUB>2</SUB>(23)