Higman–Sims group HS

The Higman–Sims group $\mathrm{HS}$ is the subgroup generated in the symmetric group by the permutations $a,b$ of $\{1,\ldots,100\}$ specified by the data files HSG1-p100B0.g1 and HSG1-p100B0.g2, respectively:

\mathrm{HS}=\langle a,b\rangle\le S_{100}.

The exact files are linked in References. Each file gives a permutation in disjoint-cycle notation; omitted points are fixed. This specifies a concrete faithful permutation model, with group multiplication given by composition.

Order and simplicity

This is a nonabelian simple group of order

|\mathrm{HS}|=44\,352\,000=2^9\cdot3^2\cdot5^3\cdot7\cdot11.

It belongs to the Leech-lattice cluster of the sporadic groups.

Geometry and neighboring groups

The action on 100 points is transitive. A point stabilizer has orbits of sizes $1,22,77$ ; the invariant graph from the orbit of length 22 is the Higman–Sims graph. Its full automorphism group is $\mathrm{HS}:2$ , so $\mathrm{HS}$ has index two in that larger group.

There is a maximal subgroup isomorphic to $\mathrm{HS}$ in $\mathrm{Co}_3$ , of index $11\,178$ .

References

ATLAS of Finite Group Representations: HS, order and representation tables.
Representation HSG1-p100B0, “About this representation” and “Checks applied”; exact GAP data generator a and generator b.
Peter J. Cameron, Scenes from mathematical life, slide “From Higman–Sims to Urysohn,” for the graph automorphism group.
ATLAS: Co3, “Maximal subgroups of Co3,” HS row.