I am currently compiling a list of known projective planes of order 49. As part of this enumeration, here are listed the plane t78 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.

Entry | Plane | |Autgp| | Point Orbits | Line Orbits | 7-rank |
---|---|---|---|---|---|

1 | Translation Plane t78, dual dt78 | 921984 | 2,4^{4},8^{2},16,2401 |
1,98,196^{4},392^{2},784 |
925 |

2 | t78_0_0, dt78_0_0 | 2058 | 1,7^{56},2058 |
1^{8},42,49^{49} |
979 |

3 | t78_0_1, dt78_0_1 | 2058 | 1,7^{56},2058 |
1^{8},42,49^{49} |
979 |

4 | t78_1_0, dt78_1_0 | 8232 | 1,7^{2},14^{9},28^{9},2058 |
1^{2},2^{3},42,49,98^{6},196^{9} |
977 |

5 | t78_1_1, dt78_1_1 | 4116 | 1,7^{8},14^{24},2058 |
1^{8},42,49,98^{24} |
979 |

6 | t78_1_2, dt78_1_2 | 8232 | 1,7^{2},14^{9},28^{9},2058 |
1^{2},2^{3},42,49,98^{6},196^{9} |
979 |

7 | t78_2_0, dt78_2_0 | 2058 | 1,7^{56},2058 |
1^{8},42,49^{49} |
981 |

8 | t78_3_0, dt78_3_0 | 4116 | 1,7^{8},14^{24},2058 |
1^{2},2^{3},42,49^{7},98^{21} |
981 |

9 | t78_4_0, dt78_4_0 | 4116 | 1,7^{8},14^{24},2058 |
1^{2},2^{3},42,49^{7},98^{21} |
981 |

10 | t78_5_0, dt78_5_0 | 4116 | 1,7^{8},14^{24},2058 |
1^{2},2^{3},42,49^{7},98^{21} |
981 |

11 | t78_6_0, dt78_6_0 | 4116 | 1,7^{8},14^{24},2058 |
1^{2},2^{3},42,49^{7},98^{21} |
981 |

12 | t78_7_0, dt78_7_0 | 8232 | 1,7^{2},14^{9},28^{9},2058 |
1^{2},2^{3},42,49,98^{6},196^{9} |
981 |

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 February, 2011