Recently Veronica Kelsey and Peter Rowley determined the M24-orbits on the 72586459 triples of octads.

Result: M24 has 16 orbits on triples of octads.

 label size label size 000;0 004;0 022;0 024;0 044;0 222;0 222;1 224;0 3795 318780 2550240 5100480 318780 10200960 4080384 7650720 224;1 224;2 244;1 244;2 444;0 444;2 444;3 444;4 20401920 2550240 6800640 7650720 35420 2550240 2266880 106260

Here the label ijk;m is for triples {X1,X2,X3} with |X1 ∩ X2| = i, |X1 ∩ X3| = j, |X2 ∩ X3| = k, |X1 ∩ X2 ∩ X3| = m. In particular, these four intersection sizes suffice to characterize the orbit.

Let us interpret this result in combinatorial terms. The octads (blocks of the Steiner system S(5,8,24)) are the vertices of a distance-transitive graph with intersection array {30,28,24; 1,3,15} and parameters given below. (See also [BCN], p. 366 and the Octad Graph.)

```v = 759 = 1 + 30 + 280 + 448    i(30,28,24; 1,3,15)
spectrum:  30**1  7**252  −3**483  −15**23
Near hexagon
Classical, b = −2
Equality in absolute bound (i,j)=(3,3)

p(1;j,k):
0:        0   1   0   0
1:        1   1  28   0
2:        0  28  28 224
3:        0   0 224 224

p(2;j,k):
0:        0   0   1   0
1:        0   3   3  24
2:        1   3 140 136
3:        0  24 136 288

p(3;j,k):
0:        0   0   0   1
1:        0   0  15  15
2:        0  15  85 180
3:        1  15 180 252

P:
1   30  280  448
1    7    4  −12
1   −3   −6    8
1  −15   70  −56
```

Graph distances 0, 1, 2, 3 correspond to octad intersection sizes 8, 0, 4, 2 and because of the triangle inequality distance triples 113, that is, intersection size triples 002, do not occur. From the p(i;j,k) we read off the number of point triples with given nonzero distances ijk.

 dist # 111 112 122 123 133 222 223 233 333 3795 318780 318780 5100480 2550240 4958800 14451360 30602880 14281344

Apparently the triples with distances 222, 223, 233, 333 split into 4, 2, 3, 2 orbits, depending on the triple intersection. Let us try to understand the corresponding geometrical situation in the near hexagon.

Near hexagon

The near hexagon has 759 points, and 3795 lines, 3 points on each line, 15 lines on each point. Any two points x, y at distance 2 determine a quad Q(x,y) that carries a subgeometry that is a generalized quadrangle of order 2, with 15 points and 15 lines. 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.

S(5,8,24)

In terms of the Steiner system S(5,8,24), the points of the near hexagon are the octads, the blocks of the design. The lines of the near hexagon are the partitions of the 24-set into three pairwise disjoint blocks. Any tetrad (4-set) determines a sextet (partition of the 24-set into six tetrads) given by the requirement that the union of any two tetrads in a sextet is an octad. The quads Q(x,y) are the sets of (6 choose 2) = 15 octads containing two tetrads of the sextet on the 4-set x∩y).

222

Look at distance 222 triples xyz. They determine three quads Q(xy), Q(xz) and Q(yz). These quads coincide precisely when |x∩y∩z| = 4 or x+y+z = 0 (with sets viewed as vectors in the extended binary Golay code). In the former case we have three points in the same ovoid (which happens for (24 choose 4).(5 choose 2) = 106260 triples xyz), in the latter case xyz is a hyperbolic line (which happens for 759.280.1/6 = 35420 triples xyz) and |x∩y∩z| = 0. If the quads do not coincide, then either x,y,z have a common neighbour w (and the three quads meet pairwise in a line) or there is no common neighbour (and the three quads meet pairwise in a single point and have empty intersection). The former happens for 759.(15.14.12/6).8 = 2550240 triples xyz (and |x∩y∩z| = 2: w is disjoint from x, y, z), the latter for 759.280.(384/6)/6 = 2266880 triples xyz (and |x∩y∩z| = 3). Check: 106260+35420+2550240+2266880 = 4958800.

223

Look at distance 223 triples xyz. There are two possibilities. Either Q(xy) and Q(xz) meet in a line L (and y and z are adjacent to distinct points y' and z' on L), or Q(xy) and Q(xz) meet in x only. The former happens for 759.15.(7 choose 2).2.4.4 = 7650720 triples xyz (and x∩y∩z = y∩z, so that |x∩y∩z| = 2), the latter for 759.35.16.8.4/2 = 6800640 triples xyz (and |x∩y∩z| = 1). Check: 7650720+6800640 = 14451360.

233

Look at distance 233 triples xyz. Let Q be the quad Q(x,y). If Q contains a neighbour w of z, then either xyw is a hyperbolic line, or x,y,w are three points of an oval. The former happens for 759.280.1.24/2 = 2550240 triples xyz (and |x∩y∩z| = 2), the latter for 759.280.3.24/2 = 7650720 triples xyz (and |x∩y∩z| = 0). If Q does not contain a neighbour of z, then z determines one of the 3 ovoids in Q not on x or y. This happens for 759.280.(384/2)/2 = 20401920 triples xyz (and |x∩y∩z| = 1). Check: 2550240+7650720+20401920 = 30602880.

333

Look at distance 333 triples xyz. The intersection size |x∩y∩z| cannot be 2 since the extended binary Golay code does not have weight 20 words. If there is a line L such that x, y, z have (distinct) neighbours on L, then |x∩y∩z| = 0. There are 3795.14.12.10.2.2.2 = 51004800 such pairs (L,xyz). Maybe there are aways 5 choices for L? Ask GAP. Yes. So, there are 51004800/5 = 10200960 such triples xyz.

If there is a quad Q such that each of x, y, z has distance 2 to Q and these points determine the same ovoid in Q, then |x∩y∩z| = 1 and there is a unique such Q. There are (24 choose 4).4.3.2.16.6.1/6 = 4080384 such triples xyz. Check: 10200960+4080384 = 14281344.

Conclusion: the 16 orbits can be distinguished geometrically. That these 16 sets of triples really are orbits follows from a permutation character computation that shows that there are just 16 orbits under M24.

label size distances geometry
000;0 3795 111 line xyz
004;0 318780 112 y,z on distinct lines on x
044;0 318780 122 x,y,z in a quad (with given distances)
024;0 5100480 123 y~x, y not in Q(xz)
022;0 2550240 133 line xyw not in quad Q(wz)
444;0 35420 222 hyperbolic line (in quad): x+y+z=0
444;2 2550240 222 x,y,z not in a quad, with common neighbour w
444;3 2266880 222 Q(xy), Q(xz), Q(yz) pairwise meet in a single point
444;4 106260 222 3 points on ovoid (in quad)
244;1 6800640 223 Q(xy), Q(xz) meet in x only
244;2 7650720 223 Q(xy), Q(xz) meet in a line
224;0 7650720 233 x,y,w on an ovoid in Q(xy), and z~w
224;1 20401920 233 Q=Q(xy), d(z,Q)=2
224;2 2550240 233 x,y,w hyperbolic line in Q(xy), and z~w
222;0 10200960 333 for some line L, d(x,L)=d(y,L)=d(z,L)=1
222;1 4080384 333 for some quad Q, x,y,z determine the same ovoid in Q

References

[BCN] A. E. Brouwer, A. M. Cohen & A. Neumaier, Distance-regular graphs, Springer Verlag, Berlin, 1989.

[BW] A. E. Brouwer & H. A. Wilbrink, The structure of near polygons with quads, Geom. Dedicata 14 (1983) 145-176.