Projective Planes of Order 49 Related to t4
I am currently compiling a list of known projective planes of order 49.
As part of this enumeration, here are listed the plane t4
and all known planes of order 49 obtained from it by dualizing and deriving. Coming soon: also
planes related by the method of lifting quotients. This list is currently incomplete; check back later
for a complete enumeration.
Following the table is a key to the table.
Known Projective Planes of Order 49 Related to t4
| Entry |
Plane |
|Autgp| |
Point Orbits |
Line Orbits |
7-rank |
| 1 |
Translation Plane t4, dual dt4 |
5531904 |
2,16,32,2401 |
1,98,784,1568 |
915 |
| 2 |
t4_0_0, dt4_0_0 |
12348 |
1,72,143,212,427,2058 |
12,6,42,49,983,1472,2946 |
975 |
| 3 |
t4_0_1, dt4_0_1 |
4116 |
1,78,1424,2058 |
12,23,42,497,9821 |
979 |
| 4 |
t4_1_0, dt4_1_0 |
6174 |
1,78,2116,2058 |
12,32,42,497,14714 |
979 |
| 5 |
t4_2_0, dt4_2_0 |
49392 |
1,282,563,168,2058 |
42,42,49,3923,1176 |
971 |
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 pzip
file containing the projective plane. I assume you have installed pzip under linux,
which compresses each plane to about 6 KB. Right-click to save as plane.pz,
then type the command punzip plane.pz
to recover plane as a text file. This text file has 2451 rows, each
row specifying a line of the plane as a subset of the points 0,1,2,...,2450.
For non-self-dual planes, a second file is also given, containing the dual
of the first.
- |Autgp| The order of the full collineation group of
the projective plane. This table entry is linked to a gzip file providing
generators of the automorphism group. Right-click to save as plane.gz,
then type the command gunzip plane.gz
to recover plane as a text
file. This text file lists generators of the full collineation group as permutations
of the integers 0,1,2,...,4901 where 0,1,2,...,2450 are labels for the points
and 2451,...,4901 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.)
- 7-rank The rank of the (0,1)-incidence matrix of the
projective plane, over a field of characteristic 7.
/
revised June, 2010