Currently there are 13 known projective planes of order 16 up to isomorphism/duality. This list includes 4 self-dual planes plus 9 dual pairs; hence 22 distinct planes up to isomorphism. Since 1997 these planes have appeared on Gordon Royle's website. I have independently verified all data appearing in the table, while retaining Royle's names for the planes, and including some additional information of interest to me concerning subplanes. The list of planes includes
Some limited attempts have been made to construct new planes of order 16 by more general net replacement and other techniques, without success. It has been conjectured that this is in fact the complete list of planes of order 16.
I have made extensive use of Brendan McKay's celebrated software package nauty for computing graph automorphisms; also the computational algebra package GAP (Graphs, Algorithms and Programming) for some of the group computations (e.g. computing conjugacy classes of involutions in groups).
If you are aware of planes which I have overlooked in my list, I would appreciate an email message () from you. For basic definitions and results on the subject of projective planes, please refer to P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, 1968; or D.R. Hughes and F.C. Piper, Projective Planes, Springer-Verlag, New York, 1973.
Following the table is a key to the table and a diagram showing connections between the planes. I have also tabulated a summary of what's known for other small orders.
I am grateful to Stefan Kranich and Martin von Gagern (TU München) for correcting my fingerprint entries in the table below.
No. | Plane | Description | 2-rank | |Autgp| | Point orbit lengths | Line orbit lengths | Derived Planes | Subplanes | Fingerprint |
---|---|---|---|---|---|---|---|---|---|
1* | desarg | Desarguesian | 82 | 17108582400 | 273 | 273 | hall | 2^{25459200} 4^{70720} | 240^{74256}256^{273} |
2* | semi2 | Semifield Plane with kernel GF(2) | 98 | 73728 | 1,16,256 | 1,16,256 | demp | 2^{2701824} 4^{1344} | 16^{2304}24^{61440}48^{2048}60^{8192}240^{272}256^{273} |
3* | semi4 | Semifield Plane with kernel GF(4) | 98 | 442368 | 1,16,256 | 1,16,256 | john, bbs4, dsfp, jowk, lmrh | 2^{5110272} 4^{10816} | 16^{2304}24^{61440}48^{2048}60^{8192}240^{272}256^{273} |
4 | hall, dhall | Hall | 98 | 921600 | 5,12,256 | 1,80,192 | desarg, hall ; bbh1, bbh2 | 2^{5644800} 4^{5440} | 240^{74256}256^{273} ; 240^{74256}256^{273} |
5 | jowk, djowk | Johnson-Walker | 100 | 258048 | 3,14,256 | 1,48,224 | semi4 ; none | 2^{5727744} 4^{8512} | 72^{13312}96^{12288}120^{43008}144^{5376}240^{272}256^{273} ; 48^{2688}96^{46592}120^{21504}144^{448}240^{3024}256^{273} |
6 | demp, ddemp | Dempwolff | 102 | 92160 | 1^{2},15,256 | 1,16^{2},240 | semi2 ; none | 2^{3010560} 4^{1888} | 96^{61440}120^{1024}144^{11520}240^{272}256^{273} ; 48^{2880}96^{53760}120^{15360}144^{480}240^{1776}256^{273} |
7 | lmrh, dlmrh | Lorimer-Rahilly | 106 | 258048 | 3,14,256 | 1,48,224 | semi4 ; none | 2^{3211776} 4^{3136} | 240^{74256}256^{273} ; 240^{74256}256^{273} |
8 | dsfp, ddsfp | Derived Semifield Plane | 106 | 55296 | 2,3,12,256 | 1,32,48,192 | semi4, dsfp ; none | 2^{3363840} 4^{3040} | 120^{6144}144^{2560}192^{61440}240^{4112}256^{273} ; 48^{960}96^{5120}120^{30720}144^{160}240^{37296}256^{273} |
9 | math, dmath | Mathon | 109 | 12288 | 1,16,256 | 1,16,256 | none ; none | 2^{3442176} 4^{2304} | 24^{6144}32^{6144}40^{24576}48^{8192}56^{12288}72^{9216}96^{3072}120^{2048}144^{2304}240^{272}256^{273} ; 24^{6144}32^{6144}40^{24576}48^{8192}56^{12288}72^{9216}96^{3072}120^{2048}144^{2304}240^{272}256^{273} |
10* | bbh1 | 110 | 9216 | 1,4,12,64,192 | 1,4,12,64,192 | dhall | 2^{3091968} 4^{832} | 0^{1152}12^{23040}24^{9216}32^{23040}40^{2304}48^{768}64^{2304}72^{6144}80^{576}96^{3072}120^{2048}144^{128}240^{464}256^{273} | |
11 | john, djohn | Johnson | 114 | 2304 | 2,3,12,16,48,96^{2} | 1,8,12^{2},24^{2},192 | semi4 ; none | 2^{2976768} 4^{784} | 0^{768}4^{1152}8^{4992}16^{3708}24^{3904}32^{9024}36^{1920}40^{4608}44^{9216}48^{5472}60^{1536}64^{9504}68^{2304}72^{768}84^{768}88^{1152}96^{864}104^{192}112^{336}120^{1920}128^{2304}132^{768}136^{2304}144^{2412}160^{960}168^{960}176^{288}208^{144}240^{8}256^{273} ; 8^{4608}12^{4608}16^{2304}24^{1152}32^{768}36^{1152}40^{2304}44^{16128}48^{5088}52^{11520}56^{7488}60^{3456}64^{1728}72^{1152}76^{2304}96^{800}112^{576}120^{2496}124^{768}136^{384}144^{64}152^{192}156^{768}240^{2448}256^{273} |
12 | bbs4, dbbs4 | 114 | 3456 | 2,3,12,16,96,144 | 1,8,12,24,36,192 | semi4 ; none | 2^{2868480} 4^{568} | 8^{3456}12^{4992}20^{6912}24^{12672}32^{7920}36^{3456}40^{288}44^{3456}48^{3312}56^{9792}64^{3888}72^{768}80^{72}84^{1152}96^{2976}104^{4032}112^{900}120^{1152}144^{1596}168^{96}192^{1056}216^{64}240^{248}256^{273} ; 8^{16128}12^{9600}24^{768}36^{2304}44^{13824}48^{11904}52^{6912}56^{1728}64^{1728}68^{1152}96^{1376}112^{288}120^{2880}136^{576}144^{88}156^{1536}240^{1464}256^{273} | |
13 | bbh2, dbbh2 | 114 | 3840 | 5,12,16,80,160 | 1,20^{2},40,192 | dhall ; dbbh2 | 2^{2866560} 4^{800} | 0^{4800}8^{640}16^{960}24^{640}36^{1280}40^{1920}48^{23040}52^{15360}60^{5504}64^{1120}68^{3840}72^{2560}80^{400}88^{640}96^{7360}100^{1920}108^{1280}144^{480}240^{512}256^{273} ; 16^{1920}20^{9216}24^{3840}36^{9600}40^{3840}44^{19200}48^{4000}60^{15744}96^{1280}112^{640}120^{2944}144^{80}240^{1952}256^{273} |
Only one line is displayed for both a plane and its dual, an asterisk (*) in the first column indicating that the plane is self-dual. Each line includes the following information and isomorphism invariants for each plane.
The image below illustrates connections between the planes we have listed. Each circle represents a dual pair of planes (or a self-dual plane). Yellow bonds between planes represent the derivability of one plane from another (a symmetric relation). Blue bonds indicate pairs of planes which share a semibiplane (also a symmetric relation).