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

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

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