McLaughlin group McL

The McLaughlin group $\mathrm{McL}$ is the subgroup generated in the symmetric group by the permutations $a,b$ of $\{1,\ldots,275\}$ specified by McLG1-p275B0.g1 and McLG1-p275B0.g2, respectively:

\mathrm{McL}=\langle a,b\rangle\le S_{275}.

The exact files are linked in References. They give disjoint-cycle permutations, with omitted points fixed, and define a faithful permutation model under composition.

Order and simplicity

This is a nonabelian simple group of order

|\mathrm{McL}|=898\,128\,000=2^7\cdot3^6\cdot5^3\cdot7\cdot11.

The 275-point action and Conway groups

The action is transitive and a point stabilizer has orbits of sizes $1,112,162$ , giving a rank-three action. The stabilizer is isomorphic to the projective special unitary group $\mathrm{PSU}_4(3)$ .

The group occurs as a maximal subgroup of $\mathrm{Co}_2$ of index $47\,104$ , and as a subgroup of $\mathrm{Co}_3$ of index $552$ . In the latter case it lies in a larger maximal subgroup $\mathrm{McL}:2$ .

References

ATLAS: McL, order and representation tables.
Representation McLG1-p275B0, “About this representation” and “Checks applied”; exact GAP data generator a and generator b.
ATLAS: Co2 and ATLAS: Co3, maximal-subgroup rows McL and McL:2, respectively.