Sporadic Suzuki group Suz

The sporadic Suzuki group $\mathrm{Suz}$ is the subgroup generated in the symmetric group by the permutations $a,b$ of $\{1,\ldots,1782\}$ specified by SuzG1-p1782B0.g1 and SuzG1-p1782B0.g2, respectively:

\mathrm{Suz}=\langle a,b\rangle\le S_{1782}.

The exact files are linked in References. They specify disjoint-cycle permutations, fixing all omitted points, and give a faithful model under composition. The symbol $\mathrm{Suz}$ denotes this single group.

Order and simplicity

This is a nonabelian simple group, with

|\mathrm{Suz}|=448\,345\,497\,600 =2^{13}\cdot3^7\cdot5^2\cdot7\cdot11\cdot13.

The 1782-point action

The action is transitive with point-stabilizer orbits of sizes $1,416,1365$ , hence has rank three. Its point stabilizer is $G_2(4)$ .

There is also a maximal subgroup $J_2:2$ , of index $370\,656$ , containing the Hall–Janko group with index two.

Naming convention

The finite simple groups of Lie type ${}^2B_2(q)$ , for $q=2^{2m+1}$ with integer $m\ge1$ , are also called Suzuki groups. That parameterized family is distinct from the sporadic group $\mathrm{Suz}$ . Likewise, the central covers $2.\mathrm{Suz}$ , $3.\mathrm{Suz}$ , and $6.\mathrm{Suz}$ are distinct groups.

References

ATLAS: Suz, order, group/cover labels, and maximal-subgroup tables.
Representation SuzG1-p1782B0, “About this representation” and “Checks applied”; exact GAP data generator a and generator b.