Generalised Polygons of Small Order

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.

These lists are provided as text files assessible through links found in the columns headed “Name” and “|Aut. Gp.|” respectively. Format: Consider a generalised polygon with M points and N lines. The first text file lists, for each point, all t+1 lines (as integers in the range 0,1,...,N−1) incident with it. The second text file lists generators of the automorphism group as permutations of {0,1,2,...,M+N−1}, in which the integers 0,1,...,M−1 represent points (in the same order as they appear in the first file) and the integers M,M+1,...,M+N−1 represent lines (in the same order as they appear in the first file, but with M added to each index).

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.

Generalised Quadrangles of Small Order

Here I have listed all known GQ's with parameters (s,t) such that st<100. The completeness of this list is known only for (s,t) equal to (2,2), (2,4), (3,3), (3,5), (3,9), (4,2), (4,4), (5,3) or (9,3). Incomplete results exist towards classifying GQ's with other small parameter sets.

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), O₅(2)	1¹⁰0⁵	720	15	15	self-dual
(2,4)	H(3,2²), U₄(2), AS(3)	1²¹0⁶	51840	27	45	dual of Q(5,2)
(3,3)	W(3), Sp(4,3)	1²⁵0¹⁵	51840	40	40	dual of Q(4,3)
(3,3)	Q(4,3), O₅(3)	1²⁵0¹⁵	51840	40	40	dual of W(3)
(3,5)	AS(4)	1⁴⁶0¹⁸	138240	64	96
(3,9)	H(3,3²), U₄(3)	1⁹⁰3¹0²¹	26127360	112	280	dual of Q(5,3)
(4,2)	Q(5,2), O₆⁻(2)	1²¹0⁶	51840	45	27	dual of H(3,2²)
(4,4)	W(4), Sp(4,4), Q(5,4), O₅(4)	1⁵⁰2¹0³⁴	1958400	85	85	self-dual
(4,6)	AS(5)	1⁸⁵0⁴⁰	60000	125	25, 150
(4,8)	H(4,2²), U₅(2)	1¹²⁰2¹0⁴⁴	27371520	165	297
(4,16)	H(3,4²), U₄(4)	1²⁶¹2¹²0⁵²	4073472000	325	1105	dual of Q(5,4)
(5,3)	O(m,n) where O is a hyperoval in PG₂(4)	1⁴⁶0¹⁸	138240	96	64	dual of AS(4)
(5,5)	W(5), Sp(4,5)	1⁹¹0⁶⁵	9360000	156	156	dual of Q(4,5)
(5,5)	Q(4,5), O₅(5)	1⁹¹0⁶⁵	9360000	156	156	dual of W(5)
(6,4)	dual AS(5)	1⁸⁵0⁴⁰	60000	25, 150	125
(6,8)	AS(7)	1²¹⁷0¹²⁶	691488	343	49, 392
(7,7)	W(7), Sp(4,7)	1²²⁵0¹⁷⁵	276595200	400	400	dual of Q(4,7)
(7,7)	Q(4,7), O₅(7)	1²²⁵0¹⁷⁵	276595200	400	400	dual of W(7)
(7,9)	AS(8)	1²⁹⁹2¹⁷0¹⁹⁶	5419008	512	64, 576	dual of O(m,n) where O is a hyperoval in PG₂(8) with nucleus n
(7,9)	dual O(m,n) where O is a hyperoval in PG₂(8) with nucleus not equal to m or n	1³⁰⁸2⁸0¹⁹⁶	150528	64, 448	16, 112, 512
(8,4)	dual H(4,2²), dual U₅(2)	1¹²⁰2¹0⁴⁴	27371520	297	165
(8,6)	dual AS(7)	1²¹⁷0¹²⁶	691488	49, 392	343
(8,8)	W(8), Sp(4,8), Q(4,8), O₅(8)	1²⁹⁸2²⁶4¹0²⁶⁰	3170119680	585	585	self-dual
(8,8)	T₂(O) where O is a nonclassical oval in PG₂(8)	1³¹⁰2¹⁴4¹0²⁶⁰	602112	1, 8, 64, 512	1, 8, 64, 512	self-dual
(8,10)	AS(9)	1⁴²⁴3¹⁷0²⁸⁸	8398080	729	81, 810	?
(9,3)	Q(5,3), O₆⁻(3)	1⁹⁰3¹0²¹	26127360	280	112	dual of H(3,3²)
(9,7)	O(m,n) where O is a hyperoval in PG₂(8) with nucleus n	1²⁹⁹2¹⁷0¹⁹⁶	5419008	64, 576	512	dual of AS(8)
(9,7)	O(m,n) where O is a hyperoval in PG₂(8) with nucleus not equal to m or n	1³⁰⁸2⁸0¹⁹⁶	150528	16, 112, 512	64, 448
(9,9)	W(9), Sp(4,9)	1⁴²⁵3²⁶0³⁶⁹	6886425600	820	820	dual of Q(4,9)
(9,9)	Q(4,9), O₅(9)	1⁴²⁵3²⁶0³⁶⁹	6886425600	820	820	dual of W(9)
(10,8)	dual AS(9)	1⁴²⁴3¹⁷0²⁸⁸	8398080	81, 810	729
(16,4)	Q(5,4), O₆⁻(4)	1²⁶¹2¹²0⁵²	4073472000	1105	325	dual of H(3,4²)

Generalised Hexagons of Small Order

Here I have listed all known GH's with parameters (s,t) such that st<40.

(s,t)	name(s)	elementary divisors	\|Aut. Gp.\|	Point orbit lengths	Line orbit lengths	Remarks
(2,2)	H(2), G₂(2)	1⁴⁹0¹⁴	12096	63	63
(2,2)	dual H(2)	1⁴⁹0¹⁴	12096	63	63
(2,8)	³D₄(2)	1⁷⁹¹2²0²⁶	634023936	819	2457
(3,3)	H(3), G₂(3)	1²⁷²3¹0⁹¹	4245696	364	364	self-dual
(4,4)	H(4), G₂(4)	1⁹⁸⁷2¹⁴0³⁶⁴	503193600	1365	1365
(4,4)	dual H(4)	1⁹⁸⁷2¹⁴0³⁶⁴	503193600	1365	1365
(5,5)	H(5), G₂(5)	1²⁷⁹⁴5²⁷0¹⁰⁸⁵	5859000000	3906	3906
(5,5)	dual H(5)	1²⁷⁹⁴5²⁷0¹⁰⁸⁵	5859000000	3906	3906
(8,2)	dual ³D₄(2)	1⁷⁹¹2²0²⁶	634023936	2457	819

Generalised Octagons of Small Order

Here I have listed the only known GO's with parameters (s,t) such that st<20.

(s,t)	name(s)	elementary divisors	\|Aut. Gp.\|	Point Orbit Lengths	Line Orbit Lengths	Remarks
(2,4)	²F₄(2)	1¹⁶⁷⁵2²0⁷⁸	35942400	1755	2925
(4,2)	dual ²F₄(2)	1¹⁶⁷⁵2²0⁷⁸	35942400	2925	1755

/ revised August, 2003