Editing Higman–Sims graph (section)

==Inside the Leech lattice==
[[Image:Higman-Sims-19.svg|220px|right|thumb|
A projection of the Higman–Sims graph inside the Leech lattice.]]
The Higman–Sims graph naturally [[Higman–Sims group#A Higman–Sims graph|occurs]] inside the [[Leech lattice]]: if ''X'', ''Y'' and ''Z'' are three points in the Leech lattice such that the distances ''XY'', ''XZ'' and ''YZ'' are <math>2, \sqrt{6}, \sqrt{6}</math> respectively, then there are exactly 100 Leech lattice points ''T'' such that all the distances ''XT'', ''YT'' and ''ZT'' are equal to 2, and if we connect two such points ''T'' and ''T''&prime; when the distance between them is <math> \sqrt{6} </math>, the resulting graph is isomorphic to the Higman–Sims graph.  Furthermore, the set of all automorphisms of the Leech lattice (that is, Euclidean congruences fixing it) which fix each of ''X'', ''Y'' and ''Z'' is the Higman–Sims group (if we allow exchanging ''X'' and ''Y'', the order 2 extension of all graph automorphisms is obtained).  This shows that the Higman–Sims group occurs inside the [[Conway group]]s Co<sub>2</sub> (with its order 2 extension) and Co<sub>3</sub>, and consequently also Co<sub>1</sub>.<ref>{{Cite book | last1=Conway | first1=John H. | authorlink1=John Horton Conway | last2=Sloane | first2=Neil J. A. | authorlink2=Neil Sloane | title=Sphere Packings, Lattices and Groups | edition=3rd | series=Grundlehren der mathematischen Wissenschaften | date=December 2010 | isbn=978-1-4419-3134-4 | publisher=[[Springer-Verlag]] }} chapter 10 (John H. Conway, "Three Lectures on Exceptional Groups"), §3.5 ("The Higman–Sims and McLaughlin groups"), pp. 292&ndash;293.</ref>