This site is intended to provide a current list of known generalised *n*-gons of small order
for *n*=4,6,8 (for *n*=3, see my page of projective
planes of small order.) We assume *s*,*t*>1.

For each generalized polygon listed, I have provided

- a list of
**point-line incidences**, and - a list of
**generators of the automorphism group**.

For basic definitions and results on the subject of generalised polygons, please refer to

- S.E. Payne, ‘A census of finite generalized quadrangles’, pp.29-36 in
*Finite Geometries, Buildings and Related Topics*, Oxford, ed. W.M. Kantor et.al., 1990. - J.A. Thas, ‘Generalized polygons’, pp.383-431 in
*Handbook of Incidence Geometry*, ed. F. Buekenhout, North-Holland, 1995. - H. Van Maldeghem, Generalized Polygons, Birkhäuser, 1998.

I have made extensive use of Brendan McKay's celebrated software package nauty for computing graph automorphisms.

If you are aware of small polygons which I have overlooked in my list, I would appreciate an email message () from you.

I am grateful to Stan Payne for informal discussions which guided me in compiling this list.

(s,t) |
name(s) | elementary divisors | |Aut. Gp.| | Point Orbit Lengths | Line Orbit Lengths | Remarks |
---|---|---|---|---|---|---|

(2,2) | W(2), Sp(4,2), Q(4,2), O_{5}(2) |
1^{10}0^{5} |
720 | 15 | 15 | self-dual |

(2,4) | H(3,2^{2}), U_{4}(2), AS(3) |
1^{21}0^{6} |
51840 | 27 | 45 | dual of Q(5,2) |

(3,3) | W(3), Sp(4,3) | 1^{25}0^{15} |
51840 | 40 | 40 | dual of Q(4,3) |

(3,3) | Q(4,3), O_{5}(3) |
1^{25}0^{15} |
51840 | 40 | 40 | dual of W(3) |

(3,5) | AS(4) | 1^{46}0^{18} |
138240 | 64 | 96 | |

(3,9) | H(3,3^{2}), U_{4}(3) |
1^{90}3^{1}0^{21} |
26127360 | 112 | 280 | dual of Q(5,3) |

(4,2) | Q(5,2), O_{6}^{−}(2) |
1^{21}0^{6} |
51840 | 45 | 27 | dual of H(3,2^{2}) |

(4,4) | W(4), Sp(4,4), Q(5,4), O_{5}(4) |
1^{50}2^{1}0^{34} |
1958400 | 85 | 85 | self-dual |

(4,6) | AS(5) | 1^{85}0^{40} |
60000 | 125 | 25, 150 | |

(4,8) | H(4,2^{2}), U_{5}(2) |
1^{120}2^{1}0^{44} |
27371520 | 165 | 297 | |

(4,16) | H(3,4^{2}), U_{4}(4) |
1^{261}2^{12}0^{52} |
4073472000 | 325 | 1105 | dual of Q(5,4) |

(5,3) | O(m,n)
where O is a hyperoval in PG_{2}(4) |
1^{46}0^{18} |
138240 | 96 | 64 | dual of AS(4) |

(5,5) | W(5), Sp(4,5) | 1^{91}0^{65} |
9360000 | 156 | 156 | dual of Q(4,5) |

(5,5) | Q(4,5), O_{5}(5) |
1^{91}0^{65} |
9360000 | 156 | 156 | dual of W(5) |

(6,4) | dual AS(5) | 1^{85}0^{40} |
60000 | 25, 150 | 125 | |

(6,8) | AS(7) | 1^{217}0^{126} |
691488 | 343 | 49, 392 | |

(7,7) | W(7), Sp(4,7) | 1^{225}0^{175} |
276595200 | 400 | 400 | dual of Q(4,7) |

(7,7) | Q(4,7), O_{5}(7) |
1^{225}0^{175} |
276595200 | 400 | 400 | dual of W(7) |

(7,9) | AS(8) | 1^{299}2^{17}0^{196} |
5419008 | 512 | 64, 576 | dual of O(m,n)
where O is a hyperoval in PG_{2}(8) with nucleus n |

(7,9) | dual O(m,n)
where O is a hyperoval
in PG_{2}(8) with nucleus not equal to m or n |
1^{308}2^{8}0^{196} |
150528 | 64, 448 | 16, 112, 512 | |

(8,4) | dual H(4,2^{2}), dual U_{5}(2) |
1^{120}2^{1}0^{44} |
27371520 | 297 | 165 | |

(8,6) | dual AS(7) | 1^{217}0^{126} |
691488 | 49, 392 | 343 | |

(8,8) | W(8), Sp(4,8), Q(4,8), O_{5}(8) |
1^{298}2^{26}4^{1}0^{260} |
3170119680 | 585 | 585 | self-dual |

(8,8) | T_{2}(O)
where O is a nonclassical oval in PG_{2}(8) |
1^{310}2^{14}4^{1}0^{260} |
602112 | 1, 8, 64, 512 | 1, 8, 64, 512 | self-dual |

(8,10) | AS(9) | 1^{424}3^{17}0^{288} |
8398080 | 729 | 81, 810 | ? |

(9,3) | Q(5,3), O_{6}^{−}(3) |
1^{90}3^{1}0^{21} |
26127360 | 280 | 112 | dual of H(3,3^{2}) |

(9,7) | O(m,n)
where O is a hyperoval
in PG_{2}(8) with nucleus n |
1^{299}2^{17}0^{196} |
5419008 | 64, 576 | 512 | dual of AS(8) |

(9,7) | O(m,n)
where O is a hyperoval
in PG_{2}(8) with nucleus not equal to m or n |
1^{308}2^{8}0^{196} |
150528 | 16, 112, 512 | 64, 448 | |

(9,9) | W(9), Sp(4,9) | 1^{425}3^{26}0^{369} |
6886425600 | 820 | 820 | dual of Q(4,9) |

(9,9) | Q(4,9), O_{5}(9) |
1^{425}3^{26}0^{369} |
6886425600 | 820 | 820 | dual of W(9) |

(10,8) | dual AS(9) | 1^{424}3^{17}0^{288} |
8398080 | 81, 810 | 729 | |

(16,4) | Q(5,4), O_{6}^{−}(4) |
1^{261}2^{12}0^{52} |
4073472000 | 1105 | 325 | dual of H(3,4^{2}) |

(s,t) |
name(s) | elementary divisors | |Aut. Gp.| | Point orbit lengths | Line orbit lengths | Remarks |
---|---|---|---|---|---|---|

(2,2) | H(2), G_{2}(2) |
1^{49}0^{14} |
12096 | 63 | 63 | |

(2,2) | dual H(2) | 1^{49}0^{14} |
12096 | 63 | 63 | |

(2,8) | ^{3}D_{4}(2) |
1^{791}2^{2}0^{26} |
634023936 | 819 | 2457 | |

(3,3) | H(3), G_{2}(3) |
1^{272}3^{1}0^{91} |
4245696 | 364 | 364 | self-dual |

(4,4) | H(4), G_{2}(4) |
1^{987}2^{14}0^{364} |
503193600 | 1365 | 1365 | |

(4,4) | dual H(4) | 1^{987}2^{14}0^{364} |
503193600 | 1365 | 1365 | |

(5,5) | H(5), G_{2}(5) |
1^{2794}5^{27}0^{1085} |
5859000000 | 3906 | 3906 | |

(5,5) | dual H(5) | 1^{2794}5^{27}0^{1085} |
5859000000 | 3906 | 3906 | |

(8,2) | dual ^{3}D_{4}(2) |
1^{791}2^{2}0^{26} |
634023936 | 2457 | 819 |

(s,t) |
name(s) | elementary divisors | |Aut. Gp.| | Point Orbit Lengths | Line Orbit Lengths | Remarks |
---|---|---|---|---|---|---|

(2,4) | ^{2}F_{4}(2) |
1^{1675}2^{2}0^{78} |
35942400 | 1755 | 2925 | |

(4,2) | dual ^{2}F_{4}(2) |
1^{1675}2^{2}0^{78} |
35942400 | 2925 | 1755 |

/ revised August, 2003