a) *A symbol*.
Γ has 2765664 21-cocliques falling in 29 orbits of sizes
22, 352, 2310, 4620, 5280, 12320, 18480, 21120, 24640, 27720,
36960 (5x), 49280 (2x), 73920, 110880 (2x), 147840 (5x),
221760 (2x), 443520 (2x).
Since 21 is the Hoffman bound on the size of a coclique, these cocliques
are maximal, and each vertex outside one has 3 neighbours inside.
For two of these 29 orbits the stabilizer of a coclique is transitive
on its 21 vertices. These are precisely the two orbits of cocliques
with a stabilizer that is maximal in Aut(Γ).
These orbits have lengths 22 and 352. A coclique in the former is called
a *symbol*, one in the latter a *heptad* (see c below).
There are 22 symbols, forming a single orbit.
The stabilizer of one is L_{3}(4).2 with vertex orbit sizes 21+210.

b) *A quad*.
Γ has 77 quads, subgeometries GQ(2,2), forming a single orbit.
The stabilizer of one is 2^{4}:S_{6}
with vertex orbit sizes 15+120+96.

c) *A heptad*.
There are 352 of these (21-cocliques, see above), forming a single orbit.
The stabilizer of one is A_{7} with vertex orbit sizes
21+105+105 (the pairs in the heptad, meeting the heptad in 1 symbol,
and those outside the heptad).

d) *A vertex*.
There are 231 of these, forming a single orbit.
The stabilizer of one is 2^{5}:S_{5}
with vertex orbit sizes 1+30+160+40.
The relation defined by the orbit of size 40 is the
triangular graph T(22).

The local graphs are 2-clique extensions of the line graph of the Petersen graph.

e) *A Fano plane*.
There are 330 of these, maximal 7-cliques on which the lines induce
a Fano plane, forming a single orbit. The stabilizer of one is
2^{3}.L_{3}(2) x 2 with vertex orbit sizes
7+28+84+112. These are the only maximal cliques.

f) *A decad / Sylvester subgraph / GO(2,1)*.
Let our S(3,6,22) be obtained by deriving S(5,8,24) twice, so that
there are two outside symbols a and b. There are 1288 splits of the
set of 24 symbols into two dodecads. Of these splits, 616 have
a and b on the same side, and 672 have them on different sides.
Let a *decad* be a set of ten symbols that together with a and b
form a dodecad.
There are 616 decads, forming a single orbit.
The stabilizer of one is A_{6}.2^{2} (of order 1440)
with vertex orbit sizes 30+36+45+120.
On the orbit of size 36 Γ induces a
Sylvester graph.
On the orbit of size 45 Γ induces a GO(2,1)
(generalized octagon of order (2,1), the flag graph of GQ(2,2),
on 45 = 1+4+8+16+16 vertices).

g) *An undecad / L2(11) subgraph*.
Let an *undecad* be a set of eleven symbols that together with
a or b form a dodecad.
There are 672 of these forming a single orbit.
The stabilizer of one is L_{2}(11):2
with vertex orbit sizes 55+66+110.
The graph induced on the orbit of size 55 has valency 6 and point
stabilizer S_{4}, see L_{2}(11)
on 55.

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

J. I. Hall & S. V. Shpectorov,
*P-geometries of rank 3*,
Geom. Dedic. **82** (2000) 139-169.