Fischer group Fi24′

The Fischer group Fi24′ $\mathrm{Fi}_{24}'$ is the subgroup generated by the fixed pair of $781\times781$ matrices $A,B$ in ATLAS dataset F24G1-f3r781B0:

\mathrm{Fi}_{24}'=\langle A,B\rangle\leq\mathrm{GL}_{781}(\mathbb F_{3}).

The files F24G1-f3r781B0.m1 and F24G1-f3r781B0.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, $\mathrm{GL}_{781}(\mathbb F_{3})$ 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

\begin{aligned}|\mathrm{Fi}_{24}'|&=2^{21}\cdot3^{16}\cdot5^{2}\cdot7^{3}\cdot11\cdot13\cdot17\\ &\quad{}\cdot23\cdot29.\end{aligned}

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 $\mathbb F_{3}^{781}$ . The matrix degree $781$ 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 prime in $\mathrm{Fi}_{24}'$ is essential: this is the simple derived subgroup of the larger Fischer group $\mathrm{Fi}_{24}=\mathrm{Fi}_{24}':2$ , which has twice its order. The central cover $3.\mathrm{Fi}_{24}'$ is also a different group.

References

ATLAS, Fischer group Fi24′, order heading and “Representations.”
ATLAS, matrix model F24G1-f3r781B0, “About this representation” and “Download”; exact matrices A and B.