Projective Planes of Order 16

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

the eight translation planes, as classified by U. Dempwolff and R. Reifart, ‘The classification of translation planes of order 16, I.’, Geometriae Dedicata 15 (1983), 137-153;
the Mathon plane; and
all planes obtained from others by duality and derivation; also by the process of lifting semibiplanes.

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.

Table of Known Projective Planes of Order 16

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 4⁷⁰⁷²⁰	240⁷⁴²⁵⁶256²⁷³
2*	semi2	Semifield Plane with kernel GF(2)	98	73728	1,16,256	1,16,256	demp	2^2701824 4¹³⁴⁴	16²³⁰⁴24⁶¹⁴⁴⁰48²⁰⁴⁸60⁸¹⁹²240²⁷²256²⁷³
3*	semi4	Semifield Plane with kernel GF(4)	98	442368	1,16,256	1,16,256	john, bbs4, dsfp, jowk, lmrh	2^5110272 4¹⁰⁸¹⁶	16²³⁰⁴24⁶¹⁴⁴⁰48²⁰⁴⁸60⁸¹⁹²240²⁷²256²⁷³
4	hall, dhall	Hall	98	921600	5,12,256	1,80,192	desarg, hall ; bbh1, bbh2	2^5644800 4⁵⁴⁴⁰	240⁷⁴²⁵⁶256²⁷³ ; 240⁷⁴²⁵⁶256²⁷³
5	jowk, djowk	Johnson-Walker	100	258048	3,14,256	1,48,224	semi4 ; none	2^5727744 4⁸⁵¹²	72¹³³¹²96¹²²⁸⁸120⁴³⁰⁰⁸144⁵³⁷⁶240²⁷²256²⁷³ ; 48²⁶⁸⁸96⁴⁶⁵⁹²120²¹⁵⁰⁴144⁴⁴⁸240³⁰²⁴256²⁷³
6	demp, ddemp	Dempwolff	102	92160	1²,15,256	1,16²,240	semi2 ; none	2^3010560 4¹⁸⁸⁸	96⁶¹⁴⁴⁰120¹⁰²⁴144¹¹⁵²⁰240²⁷²256²⁷³ ; 48²⁸⁸⁰96⁵³⁷⁶⁰120¹⁵³⁶⁰144⁴⁸⁰240¹⁷⁷⁶256²⁷³
7	lmrh, dlmrh	Lorimer-Rahilly	106	258048	3,14,256	1,48,224	semi4 ; none	2^3211776 4³¹³⁶	240⁷⁴²⁵⁶256²⁷³ ; 240⁷⁴²⁵⁶256²⁷³
8	dsfp, ddsfp	Derived Semifield Plane	106	55296	2,3,12,256	1,32,48,192	semi4, dsfp ; none	2^3363840 4³⁰⁴⁰	120⁶¹⁴⁴144²⁵⁶⁰192⁶¹⁴⁴⁰240⁴¹¹²256²⁷³ ; 48⁹⁶⁰96⁵¹²⁰120³⁰⁷²⁰144¹⁶⁰240³⁷²⁹⁶256²⁷³
9	math, dmath	Mathon	109	12288	1,16,256	1,16,256	none ; none	2^3442176 4²³⁰⁴	24⁶¹⁴⁴32⁶¹⁴⁴40²⁴⁵⁷⁶48⁸¹⁹²56¹²²⁸⁸72⁹²¹⁶96³⁰⁷²120²⁰⁴⁸144²³⁰⁴240²⁷²256²⁷³ ; 24⁶¹⁴⁴32⁶¹⁴⁴40²⁴⁵⁷⁶48⁸¹⁹²56¹²²⁸⁸72⁹²¹⁶96³⁰⁷²120²⁰⁴⁸144²³⁰⁴240²⁷²256²⁷³
10*	bbh1		110	9216	1,4,12,64,192	1,4,12,64,192	dhall	2^3091968 4⁸³²	0¹¹⁵²12²³⁰⁴⁰24⁹²¹⁶32²³⁰⁴⁰40²³⁰⁴48⁷⁶⁸64²³⁰⁴72⁶¹⁴⁴80⁵⁷⁶96³⁰⁷²120²⁰⁴⁸144¹²⁸240⁴⁶⁴256²⁷³
11	john, djohn	Johnson	114	2304	2,3,12,16,48,96²	1,8,12²,24²,192	semi4 ; none	2^2976768 4⁷⁸⁴	0⁷⁶⁸4¹¹⁵²8⁴⁹⁹²16³⁷⁰⁸24³⁹⁰⁴32⁹⁰²⁴36¹⁹²⁰40⁴⁶⁰⁸44⁹²¹⁶48⁵⁴⁷²60¹⁵³⁶64⁹⁵⁰⁴68²³⁰⁴72⁷⁶⁸84⁷⁶⁸88¹¹⁵²96⁸⁶⁴104¹⁹²112³³⁶120¹⁹²⁰128²³⁰⁴132⁷⁶⁸136²³⁰⁴144²⁴¹²160⁹⁶⁰168⁹⁶⁰176²⁸⁸208¹⁴⁴240⁸256²⁷³ ; 8⁴⁶⁰⁸12⁴⁶⁰⁸16²³⁰⁴24¹¹⁵²32⁷⁶⁸36¹¹⁵²40²³⁰⁴44¹⁶¹²⁸48⁵⁰⁸⁸52¹¹⁵²⁰56⁷⁴⁸⁸60³⁴⁵⁶64¹⁷²⁸72¹¹⁵²76²³⁰⁴96⁸⁰⁰112⁵⁷⁶120²⁴⁹⁶124⁷⁶⁸136³⁸⁴144⁶⁴152¹⁹²156⁷⁶⁸240²⁴⁴⁸256²⁷³
12	bbs4, dbbs4		114	3456	2,3,12,16,96,144	1,8,12,24,36,192	semi4 ; none	2^2868480 4⁵⁶⁸	8³⁴⁵⁶12⁴⁹⁹²20⁶⁹¹²24¹²⁶⁷²32⁷⁹²⁰36³⁴⁵⁶40²⁸⁸44³⁴⁵⁶48³³¹²56⁹⁷⁹²64³⁸⁸⁸72⁷⁶⁸80⁷²84¹¹⁵²96²⁹⁷⁶104⁴⁰³²112⁹⁰⁰120¹¹⁵²144¹⁵⁹⁶168⁹⁶192¹⁰⁵⁶216⁶⁴240²⁴⁸256²⁷³ ; 8¹⁶¹²⁸12⁹⁶⁰⁰24⁷⁶⁸36²³⁰⁴44¹³⁸²⁴48¹¹⁹⁰⁴52⁶⁹¹²56¹⁷²⁸64¹⁷²⁸68¹¹⁵²96¹³⁷⁶112²⁸⁸120²⁸⁸⁰136⁵⁷⁶144⁸⁸156¹⁵³⁶240¹⁴⁶⁴256²⁷³
13	bbh2, dbbh2		114	3840	5,12,16,80,160	1,20²,40,192	dhall ; dbbh2	2^2866560 4⁸⁰⁰	0⁴⁸⁰⁰8⁶⁴⁰16⁹⁶⁰24⁶⁴⁰36¹²⁸⁰40¹⁹²⁰48²³⁰⁴⁰52¹⁵³⁶⁰60⁵⁵⁰⁴64¹¹²⁰68³⁸⁴⁰72²⁵⁶⁰80⁴⁰⁰88⁶⁴⁰96⁷³⁶⁰100¹⁹²⁰108¹²⁸⁰144⁴⁸⁰240⁵¹²256²⁷³ ; 16¹⁹²⁰20⁹²¹⁶24³⁸⁴⁰36⁹⁶⁰⁰40³⁸⁴⁰44¹⁹²⁰⁰48⁴⁰⁰⁰60¹⁵⁷⁴⁴96¹²⁸⁰112⁶⁴⁰120²⁹⁴⁴144⁸⁰240¹⁹⁵²256²⁷³

Key to the table

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.

Plane provides a text file containing the projective plane. The text file has 273 rows, each row specifying a line of the plane as a subset of the points 0,1,2,...,272. For non-self-dual planes, a second file is also given, containing the dual of the first. The names appearing in bold face are the eight translation planes.
2-rank The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 2.
|Autgp| The order of the full automorphism group (collineation group) of the projective plane. This table entry is linked to a file providing generators of the automorphism group, listed as permutations of the integer list 0,1,2,...,545 where 0,1,2,...,272 are labels for the points and 273,...,545 are labels for the lines. Generators for the automorphism group of the dual plane are not provided since these are trivially obtained from the generators given for the original plane.
Point Orbits The lengths of the full automorphism group on the points. (These are the lengths of the line orbits for the dual plane.)
Line Orbits The lengths of the full automorphism group on the lines. (These are the lengths of the point orbits for the dual plane.)
Subplanes The orders of the proper subplanes, and the total number of each.
Fingerprint The fingerprint of each plane (an isomorphism invariant) is provided. For dual pairs of planes, the fingerprint of the first plane in the pair is given; then a colon, followed by the fingerprint of the second plane in the pair. Refer to the accompanying definition of the fingerprint.

Connections between the planes

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).

/ revised May, 2010