This point-line geometry is a *near polygon*: for each point x and each
line L there is a unique point on L closest (in graph distance) to x.
More precisely, it is a *near hexagon*, since Γ has diameter 3.
Given a set S of points, let C(S) be the geodetic closure of S,
that is, the smallest subspace containing S that is closed for geodetics.
If x, y are points with d(x,y) = 2, then C({x,y}) is a subgeometry
with 15 points and 15 lines called a *quad*.
It is a generalized quadrangle of order 2.
The 15 lines and 35 quads on a fixed point x form the points and lines
of a geometry PG(3,2).
If Q is a quad, and z a point with d(z,Q)=2, then z has distance 2
to five points of Q, and these five points form an ovoid in Q.
See also [BCN], §11.4A.

Let ω be a fixed element of Ω. There are 253 octads that contain ω, and 506 that do not contain ω. The graph Δ is the induced subgraph in Γ on the set of 506. The set of 253 is a big ovoid O: a set that meets every line of the near polygon in precisely one point. (Above we saw 5-point ovoids in a quad. These are the intersections of a big ovoid and a quad. Also the intersections of four big ovoids.) There are no other big ovoids than the 24 obtained in this way.

The 24 big ovoids are permuted by M_{24}. The stabilizer of one
is M_{23}, with vertex stabilizer 2^{4}:A_{7}.

The full automorphism group of Δ is M_{23},
with vertex stabilizer A_{8}.
The graph Δ is distance-transitive.

The intersection of Δ with a quad is a Petersen graph. For each vertex x of Δ, the 15 edges and 35 Petersen graphs and 15 symbols (in Ω \ (x ∪ {ω})) form the points and lines and planes of a geometry PG(3,2). See also [BCN], §11.4B.

Let σ, τ be two fixed elements of Ω.
There are 77 octads that contain both σ and τ,
352 that contain precisely one, and 330 that contain neither.
The graph E is the induced subgraph in Γ on the set of 330.
The full automorphism group of E is M_{22}.2, with point stabilizer
2^{3}:L_{3}(2)×2.
The graph E is distance-transitive. See also [BCN], §11.4C.

A. E. Brouwer,
*The uniqueness of the near hexagon on 759 points*,
pp 47-60 in: Finite Geometries, T. G. Ostrom Conf. Pullman 1981,
Lecture Notes in Pure and Applied Math. **82**, Marcel Dekker,
New York, 1983.

A. E. Brouwer,
*Uniqueness and nonexistence of some graphs related to M _{22}*,
Graphs Combin.

A. E. Brouwer, A. M. Cohen & A. Neumaier,
*Distance-regular graphs*,
Springer, Heidelberg, 1989.

A. E. Brouwer & E. W. Lambeck,
*An inequality on the parameters of distance regular graphs
and the uniqueness of a graph associated to M _{23}*,
Ann. Discrete Math.

J. H. Conway,
*Three lectures on exceptional groups*,
pp 215-247 in: Finite Simple Groups,
Academic Press, 1971.

E. E. Shult & A. Yanushka,
*Near n-gons and line systems*,
Geom. Dedicata **9** (1980) 1-72.