a) *A vertex*.

There are 100 of these, forming a single orbit.
The stabilizer of one (in the full automorphism group)
is M_{22}.2 with vertex orbit sizes 1+22+77.
The subgraph induced on the second subconstituent is
the M22 graph, the unique
strongly regular graph with v=77, k=16, λ=0.

b) *A split into two Hoffman-Singleton graphs*.

There are 352 of these, forming a single orbit.
The stabilizer is U_{3}(5).2 with vertex orbit size 100.
(That such splits exist is shown in the description of the
Hoffman-Singleton graph,
where the Higman-Sims graph is constructed on the 100
15-cocliques of the Hoffman-Singleton graph.)

Under HS these 352 fall into two families of size 176, and HS acts doubly transitively on both. This is the Higman geometry of 176 points and 176 quadrics.

The 704 Hoffman-Singleton subgraphs of Γ meet in 0,50 (1x), 15,35 (50x), 20,30 (175x), or 25,25 (126x) points. The point-quadric incidence in Higman's geometry corresponds to the 15,35,15,35 intersection of the corresponding splits.

c) *An edge*.

There are 1100 of these, forming a single orbit.
The stabilizer of one is L_{3}(4):2^{2}
with vertex orbit sizes 2+42+56.
The subgraph induced on the 42 is the point-line incidence graph
of the projective plane PG(2,4).
The subgraph induced on the 56 is the
Gewirtz graph.

d) *A PG(3,2) point-plane nonincidence graph*

As we saw, Γ contains the bipartite point-plane nonincidence
graph of PG(3,2). There are 1100 of these, forming a single orbit.
The stabilizer of one is S_{8}×2 with vertex orbit sizes 30+70.
The subgraph induced on the 30 is our point-plane nonincidence graph
of PG(3,2). It is distance-regular with intersection array {8,7,4;1,4,8}.
The subgraph induced on the 70 is the graph on the 4-subsets of an 8-set,
adjacent when they meet in a single element.

Let us describe this in terms of the Steiner system
S(5,8,24). Pick two symbols *a*, *b*, and let S(3,6,22)
consist of the 77 octads on *a*,*b* (with these elements removed).
Pick an octad B not containing *a*,*b*.
The 77 blocks of S(3,6,22) now split 14+56+7 for having 4,2,0 symbols
in common with B, and in the 1+22+77 construction of Γ,
the (1+14)+(8+7) induces the point-plane nonincidence graph of PG(3,2).

If we want to extend this 15+15 to a split into 50+50 defined by
some octad C containg *a* but not *b*, then C must be
disjoint from B, and we have 8 choices. And if {B,C,D} is a partition
of the set of 24 symbols, then D defines the complementary split
(the 35's switch sides).

The objects we found here are known as *conics* in Higman's geometry.
They are K_{8,8}'s in the point-quadric incidence graph:
sets of 8 points contained in 8 quadrics. Each pair of points is in
precisely two conics, and their union is contained in two quadrics.
Dually, the intersection of two quadrics (of size 14) contains two conics
that meet in two points. The conic B defines a polarity by sending
the point C to the quadric D when {B,C,D} is as above.

e) *A non-edge*.

There are 3850 of these, forming a single orbit.
The stabilizer of one is 2^{5}.S_{6}
with vertex orbit sizes 2+6+32+60.
The subgraph induced on the 32 is the folded 6-cube.
The subgraph induced on the 60 is the second subconstituent
of the M22 graph.

g) *2-coclique extension of the Petersen graph*.

There are 5775 of these, forming a single orbit.
The stabilizer of one is 2_{+}^{1+6}:S_{5}
with vertex orbit sizes 20+80.
The subgraph induced on the 20 is the 2-coclique extension of the
Petersen graph. (The one on 80 is messy.)

i) *Pair of splits from the same family*.

In terms of Higman's geometry: a pair of points.
(Note that a pair of points determines a pair of quadrics
and vice versa.)
There are 15400 of these, forming a single orbit.
The stabilizer of one is (2×A_{6}.2^{2}).2
with vertex orbit sizes ***40+60***

j) *Partitions into four 5C _{5}*.

As we saw, for each of the 176 50+50 splits from one family there are 126 from the other family such that the common refinement is 25+25+25+25. Thus, there are 176*126 = 22176 such unordered pairs of splits, and these form a single orbit. The stabilizer of one is 5

k) *Good partitions into ten {4,3,1;1,3,4}*.

Γ contains 61600 subgraphs isomorphic to
K_{5,5} minus a matching, the complement of the 2×5 grid,
the unique distance-regular graph with intersection array {4,3,1;1,3,4},
the bipartite double of the complete graph K_{5}.
For brevity, let us call such a graph a BD(K_{5}).
These subgraphs form a single orbit, with stabilizer
6.A_{5}.2^{2} of order 1440 and vertex orbit sizes 10+30+60.
Counting is easy in the 1+22+77 construction: such a subgraph containing
the 1 is uniquely determined by 4 symbols not in a block (and we find the
four blocks that contain 3 of the 4 symbols, and the unique block
disjoint from the previous four blocks).

On the 30 the induced subgraph is the point-plane nonincidence graph
of PG(3,2). Now this graph has a unique split (up to isomorphism)
into three BD(K_{5})'s, since the point-plane
incidence graph has a unique split into three 5K_{2}'s.
(Description: Let PG(3,2) have points and hyperplanes given by the
nonzero elements of GF(16), where x is incident with y when
Tr(xy) = 0. Since the only cube of trace 0 is 1, partitioning
points and planes according to exponent mod 3 works.)

Our object is a partition of Γ into ten BD(K_{5})'s
where this set of ten carries the structure of a Petersen graph,
and for each BD(K_{5}) the subgraph induced on the union of
its three neighbours is the point-plane nonincidence graph of PG(3,2).
There are 36960 *** of these, forming a single orbit.
The stabilizer of one is 5:4 × S_{5}
with vertex orbit size 100.

This description is a bit involved, but just saying
"Partition into ten BD(K_{5})'s" is not enough:
Γ has a very large number of such partitions.

A.E. Brouwer,
*Polarities of G. Higman's symmetric design and a strongly regular
graph on 176 vertices*,
Aequationes Math. **25** (1982) 77-82.

A. Gewirtz,
*Graphs with maximal even girth*,
Canad. J. Math. **21** (1969) 915-934.

D.G. Higman & C. Sims,
*A simple group of order 44,352,000*,
Math.Z. **105** (1968) 110-113.

G. Higman,
*On the simple group of D.G. Higman and C.C. Sims*,
Illinois J. Math. **13** (1969) 74-80.

D.M. Mesner,
*An investigation of certain combinatorial properties of partially
balanced incomplete block experimental designs and association schemes,
with a detailed study of designs of Latin square and related types*,
Ph.D. Thesis, Michigan State University, 1956.
(Local copy here.)

D.M. Mesner,
*Negative Latin square designs*,
Institute
of Statistics, UNC, NC Mimeo series **410**, November 1964.

M.S. Smith,
*On the isomorphism of two simple groups of order 44,352,000*,
J. Algebra **41** (1976) 172-174.