Projective Planes of Order 27

This site is intended to provide a current list of known projective planes of order 27. I have listed the 13 planes of which I am aware (3 self-dual planes plus 5 dual pairs). These include

the seven translation planes, as classified by U. Dempwolff, ‘Translation planes of order 27’, Designs, Codes and Cryptography 27 (1994), 105-121 (denoted here by I,...,VII following Dempwolff);
the duals of translation planes III-VII; and
the Figueroa plane; see e.g. A. Beutelspacher, ‘Projective planes’, pp.107-136 in Handbook of Incidence Geometry, ed. F. Buekenhout, North-Holland, 1995.

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.

I have verified (on October 21, 2008) that all planes in this list, except for the Desarguesian plane desarg, have subplanes of order 2. (It has been conjectured that all finite projective planes, other than Desarguesian planes of odd order, have subplanes of order 2.)

Following the table is a key to the table. I have also tabulated a summary of what's known for other small orders.

Table of Known Projective Planes of Order 27

No.	Plane	Description	3-rank	\|Autgp\|	Point orbit lengths	Line orbit lengths	Subplanes	Fingerprint
I*	desarg	Desarguesian	217	846083360304	757	757	3^50218623	0⁵⁷²²⁹²729⁷⁵⁷
II*	twisted	Generalized twisted field plane	262	3070548	1,27,729	1,27,729	2^65505024 3^1601613	0⁵⁷²²⁹²729⁷⁵⁷
III	hering, heringD	Hering	274	1592136	28,729	1,756	2^59363928 3⁸⁶²⁴⁰⁷	0⁵⁷²²⁹²729⁷⁵⁷; 0⁵⁷²²⁹²729⁷⁵⁷
IV	flag4, flag4D	Flag-transitive	271	122472	28,729	1,756	2^68689296 3⁸⁶²⁴⁰⁷	0⁵⁷²²⁹²729⁷⁵⁷; 0⁵⁷²²⁹²729⁷⁵⁷
V	sherk, sherkD	Sherk	273	118098	1,27,729	1,27,729	2^66174246 3⁸⁶²⁴⁰⁷	0⁶⁰²¹⁰16³⁵⁴²⁹⁴24³⁹³⁶⁶56¹¹⁸⁰⁹⁸108³²⁴729⁷⁵⁷; 0³²⁶⁶⁴⁶4²³⁶¹⁹⁶216⁹²³⁴432¹⁶²648⁵⁴729⁷⁵⁷
VI	flag6, flag6D	Flag-transitive	265	122472	28,729	1,756	2^69587424 3⁸⁶²⁴⁰⁷	0⁴¹²⁴⁴8¹²²⁴⁷²24¹²²⁴⁷²32²⁴⁴⁹⁴⁴48⁴⁰⁸²⁴108¹⁶⁸216¹⁶⁸729⁷⁵⁷; 0³¹⁸²⁷⁶4¹²²⁴⁷²8¹²²⁴⁷²216⁹⁰⁷²729⁷⁵⁷
VII	andre, andreD	Andre	268	1478412	2,26,729	1,54,702	2^51744420 3⁹³⁸²²³	0⁵⁷¹⁸⁷⁶324²⁶⁰432¹⁵⁶729⁷⁵⁷; 0²⁶⁹⁰²⁸12¹⁸⁹⁵⁴⁰16¹¹³⁷²⁴729⁷⁵⁷
VIII*	fig	Figueroa	328	16848	13,312,432	13,312,432	2^58608576 3¹²³⁹⁰³	0¹⁸⁸⁷⁶4¹⁰¹⁰⁸⁸16⁶⁷³⁹²24⁶⁷³⁹²28³³⁶⁹⁶32¹⁶⁸⁴⁸36¹¹²³²40⁸⁹⁸⁵⁶44³³⁶⁹⁶48⁵⁶¹⁶⁰56³³⁶⁹⁶60¹¹²³²72¹⁸⁷²80¹⁶⁸⁴⁸144¹⁸⁷²168⁵⁶¹⁶208⁸⁶⁴216³⁴³²432⁶²⁴729⁷⁵⁷

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 757 rows, each row specifying a line of the plane as a subset of the points 0,1,2,...,756. For non-self-dual planes, a second file is also given, containing the dual of the first.
3-rank The rank of the (0,1)-incidence matrix of the projective plane, over a field of characteristic 3.
|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,...,1513 where 0,1,2,...,756 are labels for the points and 757,...,1513 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.

/ revised September, 2000