Result: M_{24} has 16 orbits on triples of octads.

label | 000;0 | 004;0 | 022;0 | 024;0 | 044;0 | 222;0 | 222;1 | 224;0 |
---|---|---|---|---|---|---|---|---|

size | 3795 | 318780 | 2550240 | 5100480 | 318780 | 10200960 | 4080384 | 7650720 |

label | 224;1 | 224;2 | 244;1 | 244;2 | 444;0 | 444;2 | 444;3 | 444;4 |

size | 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.

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 M_{24}.

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 |

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