Fischer group Fi23 - Knowlpedia

The Fischer group Fi23 $\mathrm{Fi}_{23}$ is the subgroup generated by the fixed pair of $253\times253$ matrices $A,B$ in ATLAS dataset F23G1-f3r253B0:

\mathrm{Fi}_{23}=\langle A,B\rangle\leq\mathrm{GL}_{253}(\mathbb F_{3}).

The files F23G1-f3r253B0.m1 and F23G1-f3r253B0.m2 specify $A$ and $B$ , respectively; their entries are part of this concrete definition. Here $\mathbb F_{3}=\mathbb Z/3\mathbb Z$ is the field with 3 elements, $\mathrm{GL}_{253}(\mathbb F_{3})$ consists of invertible matrices, and the group operation is matrix multiplication over that field. The exact generator files are linked in References.

Order and structure

This is a nonabelian finite simple group, with

|\mathrm{Fi}_{23}|=4089470473293004800=2^{18}\cdot3^{13}\cdot5^{2}\cdot7\cdot11\cdot13\cdot17\cdot23.

It belongs to the Monster display block of the sporadic groups. This grouping records the ATLAS organization, without asserting a direct embedding into another displayed group.

The concrete matrix model

This gives a faithful action on $\mathbb F_{3}^{253}$ . The matrix degree $253$ is distinct from the number of group elements. The two specified matrices determine the group; tests on the orders of unspecified generators would not constitute this definition.

The sporadic subgroups of the Baby Monster include copies of $\mathrm{Fi}_{23}$ and $\mathrm{Th}$ .

References

ATLAS, Fischer group Fi23, order heading and “Representations.”
ATLAS, matrix model F23G1-f3r253B0, “About this representation” and “Download”; exact matrices A and B.