The Lattice Gamma72
An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, RWTH Aachen University
(nebe@math.rwth-aachen.de)
and
Neil J. A. Sloane
(njasloane@gmail.com)
Last modified Fri Jul 18 13:28:23 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DET
MINIMAL_NORM
KISSING_NUMBER
SUBGROUP_ORDER
SUBGROUP
SUBGROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
LAST_LINE
-
NAME
Gamma72
-
DIMENSION
72
-
GRAM
72
8 0 0 -4 -4 4 -4 0 -2 4 -4 -2 -4 -2 4 2 -4 4 -4 4 2 -2 -4 -2 -2 -1 0 1 0 -2 2
-1 2 -1 1 2 1 1 0 0 2 1 3 -3 0 1 1 0 2 -1 0 -1 -2 0 0 -1 1 1 -1 1 -1 0 2 1 0
3 1 -1 1 0 -1 -1
0 8 -4 2 2 2 0 4 -2 0 4 2 4 -4 -2 0 -4 -4 -2 4 -4 4 0 -4 1 -2 2 -2 0 1 -1 -1 1
1 -2 0 -1 0 0 0 -1 2 1 -1 1 0 1 1 1 2 0 -1 1 2 -1 1 0 1 0 1 1 -2 -1 0 -3 0 0
1 -1 2 1 -1
0 -4 8 -4 2 2 0 -4 4 2 -2 2 -4 2 4 -2 2 2 4 -2 4 -2 0 2 0 0 -2 3 -2 -1 -1 0 -2
-1 1 -3 1 0 0 -1 1 -1 -1 0 -1 1 -2 0 0 -2 2 1 -1 0 -1 -2 0 0 0 -2 -1 1 2 -2
2 0 1 -1 1 0 -2 1
-4 2 -4 8 0 -2 0 0 -2 -4 4 -2 4 -2 -4 -2 0 -4 0 -2 -4 4 0 -2 1 1 -1 -2 1 0 1 2
-1 0 -1 0 0 0 -2 2 -1 0 -2 2 -1 -2 1 1 -1 2 -3 2 1 -1 1 2 -2 -2 1 -1 2 -1 -4
1 -1 -2 -2 1 -3 0 1 0
-4 2 2 0 8 0 4 0 4 -2 2 4 2 2 -2 0 0 -4 4 0 -2 0 2 2 2 -1 1 -1 -2 2 -2 0 -1 3
-2 -2 -1 0 1 -2 -1 1 -2 1 1 0 -2 0 0 0 2 -1 2 2 0 0 1 2 -1 0 0 1 0 -2 -1 -1
0 1 0 0 -1 1
4 2 2 -2 0 8 -4 0 0 4 -2 0 -2 -4 2 -2 -4 2 -2 4 2 2 -4 -4 0 -2 0 1 -2 -2 0 -1 0
0 0 -1 0 1 0 0 1 1 1 -2 -1 0 0 0 2 -1 1 0 -2 2 -2 -1 0 2 -1 -1 -1 -1 1 -1 -1
2 0 0 0 1 -2 -2
-4 0 0 0 4 -4 8 2 4 -2 0 4 2 4 -2 2 4 -2 2 -2 -2 -2 2 4 0 1 1 -1 0 2 -2 1 -1 2
0 -1 0 -1 1 -2 -1 -1 -1 2 1 1 0 0 -2 1 1 -1 2 0 2 2 1 1 0 1 1 1 0 -1 1 -2 0
1 0 0 1 2
0 4 -4 0 0 0 2 8 0 2 0 0 4 0 -2 0 0 -2 -4 4 -2 0 2 0 1 -1 2 -2 0 1 -2 -2 0 1 -1
0 -2 -1 2 1 -2 1 1 -1 1 0 1 0 1 1 0 -2 0 1 -1 2 0 2 -1 0 0 -1 1 1 -2 0 -1 1
0 0 2 0
-2 -2 4 -2 4 0 4 0 8 2 -2 4 0 4 2 -2 2 0 4 -2 2 -2 2 2 -1 0 0 2 -1 0 -1 0 -2 0
1 -2 1 -1 0 -2 0 -1 -1 0 -1 1 -1 1 -2 -1 2 1 1 0 1 0 2 1 0 0 1 1 1 -3 1 -1 1
-1 0 0 0 2
4 0 2 -4 -2 4 -2 2 2 8 -4 0 -2 0 4 -2 -2 2 -2 2 2 -2 -2 -2 -1 -1 0 2 -2 -2 -1
-2 -1 -2 2 -1 1 -1 1 0 1 0 1 -3 0 2 1 0 1 -1 1 0 -3 0 -2 -1 0 2 0 -1 0 -1 3
-1 0 1 0 -2 1 1 0 -1
-4 4 -2 4 2 -2 0 0 -2 -4 8 2 4 -2 -2 -2 0 -4 2 0 -2 4 2 -2 1 0 0 -1 1 1 0 1 0 0
-2 0 -1 0 -2 1 -1 0 -1 1 0 -1 1 1 -1 2 -1 1 2 0 0 1 -1 -2 2 1 1 -1 -3 0 -1
-2 0 1 -1 1 2 0
-2 2 2 -2 4 0 4 0 4 0 2 8 0 0 0 0 0 0 4 0 0 0 2 0 -1 -1 2 1 0 1 -1 0 0 1 -1 -2
0 0 0 -2 1 -1 0 -1 0 1 0 1 -2 0 3 0 2 1 1 0 2 1 0 2 0 0 0 -2 1 -1 2 -1 0 1 1
1
-4 4 -4 4 2 -2 2 4 0 -2 4 0 8 0 -4 -2 0 -4 0 0 -4 4 2 0 1 -1 1 -2 0 1 -1 0 -1 0
-1 0 -2 -1 0 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 0 1 0 0 2 -1 -1 1 0 2 -1 -2 0 -1
-1 -1 1 -1 1 2 1
-2 -4 2 -2 2 -4 4 0 4 0 -2 0 0 8 0 0 4 0 2 -2 2 -4 2 4 0 2 -1 1 -1 1 -1 1 -1 1
1 0 1 -2 1 -2 -1 -1 -1 1 0 1 -1 -1 -1 0 0 0 0 -1 1 1 1 1 0 0 1 2 1 -2 1 -1 0
0 1 -1 0 1
4 -2 4 -4 -2 2 -2 -2 2 4 -2 0 -4 0 8 0 0 2 0 0 4 -2 -2 0 -2 1 -2 4 0 -1 0 -1 -1
-3 3 0 2 -1 -2 -1 0 -1 1 -1 -1 3 0 -1 0 0 0 2 -1 0 -1 -2 0 -1 2 0 0 -1 2 -1
0 0 1 -1 1 2 -1 -1
2 0 -2 -2 0 -2 2 0 -2 -2 -2 0 -2 0 0 8 0 0 -2 0 -2 -2 0 2 -1 0 2 -1 2 1 1 -1 3
1 0 2 0 2 1 -2 0 0 2 0 2 1 0 -1 0 0 1 -2 2 0 2 -1 2 0 -1 2 -1 2 1 2 0 0 1 0
1 0 0 0
-4 -4 2 0 0 -4 4 0 2 -2 0 0 0 4 0 0 8 0 2 -4 2 -2 2 4 0 3 -2 1 1 1 -1 2 -1 0 1
-1 1 -1 0 0 -2 -3 -2 3 -1 1 0 -1 -2 1 -1 1 1 -1 1 2 0 -1 1 -1 1 1 0 0 2 -3
-1 1 0 0 1 1
4 -4 2 -4 -4 2 -2 -2 0 2 -4 0 -4 0 2 0 0 8 0 0 4 -2 -2 0 -3 0 0 2 1 -2 2 0 1 -1
2 1 1 1 0 0 3 -2 2 -2 -1 0 0 0 -1 -2 1 0 -1 -1 1 -1 1 0 0 1 -1 1 1 0 3 2 2
-2 1 -1 -1 0
-4 -2 4 0 4 -2 2 -4 4 -2 2 4 0 2 0 -2 2 0 8 -4 0 0 2 2 -1 0 -1 2 0 0 0 1 -1 0 0
-2 1 0 -1 -1 1 -2 -2 0 -1 0 -1 1 -3 -1 1 2 2 -1 1 -1 1 -1 1 0 1 1 -1 -2 2 -2
2 -2 -1 0 0 2
4 4 -2 -2 0 4 -2 4 -2 2 0 0 0 -2 0 0 -4 0 -4 8 0 0 -2 -2 1 -1 1 -1 -1 0 -1 -1 1
2 -1 1 -1 0 1 0 -1 2 2 -2 0 0 1 -1 3 1 0 -2 -1 2 -2 1 0 3 -1 1 -1 -1 1 0 -3
2 0 2 0 0 0 -2
2 -4 4 -4 -2 2 -2 -2 2 2 -2 0 -4 2 4 -2 2 4 0 0 8 -2 0 0 -1 1 -1 3 0 0 0 0 0 -1
1 0 1 -1 -1 -1 0 -1 1 0 -2 1 -1 -1 0 -1 1 1 -1 1 -1 -1 1 0 0 0 -1 0 1 -2 1 1
1 0 2 0 -1 -1
-2 4 -2 4 0 2 -2 0 -2 -2 4 0 4 -4 -2 -2 -2 -2 0 0 -2 8 0 -4 0 -2 0 0 0 -1 0 0 0
-1 -1 -1 -1 1 -2 0 0 1 0 0 0 -2 1 1 -1 0 -1 2 0 0 -1 0 -1 -2 1 -1 1 -1 -3 -1
-1 0 0 0 -1 2 1 -1
-4 0 0 0 2 -4 2 2 2 -2 2 2 2 2 -2 0 2 -2 2 -2 0 0 8 2 1 -1 2 -1 1 2 -1 -2 0 0
-2 -1 -2 0 1 0 -1 1 0 0 1 -1 -2 1 -1 -1 2 -1 2 0 0 -1 1 -1 -1 0 -1 1 0 0 0 0
1 -1 1 -1 2 2
-2 -4 2 -2 2 -4 4 0 2 -2 -2 0 0 4 0 2 4 0 2 -2 0 -4 2 8 1 1 -1 0 -1 2 -2 0 -2 1
0 -1 -1 -1 1 0 -1 0 -2 2 1 1 -2 -2 0 -1 0 -1 0 0 0 0 -1 0 -1 -1 -1 1 1 1 1 0
-1 1 1 -1 -1 2
-2 1 0 1 2 0 0 1 -1 -1 1 -1 1 0 -2 -1 0 -3 -1 1 -1 0 1 1 8 0 0 -4 -4 4 -4 0 -2
4 -4 -2 -4 -2 4 2 -4 4 -4 4 2 -2 -4 -2 2 1 0 -1 0 2 -2 1 -2 1 -1 -2 -1 -1 0
0 -2 -1 -3 3 0 -1 -1 0
-1 -2 0 1 -1 -2 1 -1 0 -1 0 -1 -1 2 1 0 3 0 0 -1 1 -2 -1 1 0 8 -4 2 2 2 0 4 -2
0 4 2 4 -4 -2 0 -4 -4 -2 4 -4 4 0 -4 -1 2 -2 2 0 -1 1 1 -1 -1 2 0 1 0 0 0 1
-2 -1 1 -1 0 -1 -1
0 2 -2 -1 1 0 1 2 0 0 0 2 1 -1 -2 2 -2 0 -1 1 -1 0 2 -1 0 -4 8 -4 2 2 0 -4 4 2
-2 2 -4 2 4 -2 2 2 4 -2 4 -2 0 2 0 0 2 -3 2 1 1 0 2 1 -1 3 -1 0 0 1 -1 1 1 0
1 -1 2 0
1 -2 3 -2 -1 1 -1 -2 2 2 -1 1 -2 1 4 -1 1 2 2 -1 3 0 -1 0 -4 2 -4 8 0 -2 0 0 -2
-4 4 -2 4 -2 -4 -2 0 -4 0 -2 -4 4 0 -2 -1 -1 1 2 -1 0 -1 -2 1 0 1 0 0 0 2 -2
1 0 2 -2 1 2 -1 -1
0 0 -2 1 -2 -2 0 0 -1 -2 1 0 0 -1 0 2 1 1 0 -1 0 0 1 -1 -4 2 2 0 8 0 4 0 4 -2 2
4 2 2 -2 0 0 -4 4 0 -2 0 2 2 -2 1 -1 1 2 -2 2 0 1 -3 2 2 1 0 -1 2 1 -1 2 -1
-1 0 2 0
-2 1 -1 0 2 -2 2 1 0 -2 1 1 1 1 -1 1 1 -2 0 0 0 -1 2 2 4 2 2 -2 0 8 -4 0 0 4 -2
0 -2 -4 2 -2 -4 2 -2 4 2 2 -4 -4 0 2 0 -1 2 2 0 1 0 0 0 1 0 -1 0 0 -1 -1 -1
2 1 0 0 0
2 -1 -1 1 -2 0 -2 -2 -1 -1 0 -1 -1 -1 0 1 -1 2 0 -1 0 0 -1 -2 -4 0 0 0 4 -4 8 2
4 -2 0 4 2 4 -2 2 4 -2 2 -2 -2 -2 2 4 0 -1 -1 1 0 -2 2 -1 1 -2 0 1 0 1 -1 2
1 1 1 -2 -1 -1 0 0
-1 -1 0 2 0 -1 1 -2 0 -2 1 0 0 1 -1 -1 2 0 1 -1 0 0 -2 0 0 4 -4 0 0 0 2 8 0 2 0
0 4 0 -2 0 0 -2 -4 4 -2 0 2 0 -1 1 -2 2 0 -1 2 2 0 -1 1 0 2 1 -2 -1 2 -1 -1
1 -1 0 -1 0
2 1 -2 -1 -1 0 -1 0 -2 -1 0 0 -1 -1 -1 3 -1 1 -1 1 0 0 0 -2 -2 -2 4 -2 4 0 4 0
8 2 -2 4 0 4 2 -2 2 0 4 -2 2 -2 2 2 1 0 0 -2 1 0 1 0 2 0 -1 2 -1 1 0 2 0 1 1
0 1 -1 1 -1
-1 1 -1 0 3 0 2 1 0 -2 0 1 0 1 -3 1 0 -1 0 2 -1 -1 0 1 4 0 2 -4 -2 4 -2 2 2 8
-4 0 -2 0 4 -2 -2 2 -2 2 2 -2 -2 -2 1 1 0 -2 2 2 1 2 1 2 -2 1 -1 1 -1 0 -1 0
-1 3 0 -2 -1 0
1 -2 1 -1 -2 0 0 -1 1 2 -2 -1 -1 1 3 0 1 2 0 -1 1 -1 -2 0 -4 4 -2 4 2 -2 0 0 -2
-4 8 2 4 -2 -2 -2 0 -4 2 0 -2 4 2 -2 -1 0 0 1 -1 -1 0 -1 0 0 2 0 1 0 2 -1 1
0 1 -1 0 1 -1 -1
2 0 -3 0 -2 -1 -1 0 -2 -1 0 -2 0 0 0 2 -1 1 -2 1 0 -1 -1 -1 -2 2 2 -2 4 0 4 0 4
0 2 8 0 0 0 0 0 0 4 0 0 0 2 0 1 1 -2 -1 0 -1 1 0 0 -1 1 2 0 0 0 2 -1 1 0 1 0
-1 0 -1
1 -1 1 0 -1 0 0 -2 1 1 -1 0 -2 1 2 0 1 1 1 -1 1 -1 -2 -1 -4 4 -4 4 2 -2 2 4 0
-2 4 0 8 0 -4 -2 0 -4 0 0 -4 4 2 0 -1 1 -1 2 0 -1 1 0 1 0 1 0 2 1 0 -1 1 -1
1 -1 -1 1 -1 -1
1 0 0 0 0 1 -1 -1 -1 -1 0 0 -1 -2 -1 2 -1 1 0 0 -1 1 0 -1 -2 -4 2 -2 2 -4 4 0 4
0 -2 0 0 8 0 0 4 0 2 -2 2 -4 2 4 0 -2 1 -1 1 -1 1 -1 1 -1 -1 0 -1 2 -1 2 1 1
1 -1 0 -1 1 1
0 0 0 -2 1 0 1 2 0 1 -2 0 0 1 -2 1 0 0 -1 1 -1 -2 1 1 4 -2 4 -4 -2 2 -2 -2 2 4
-2 0 -4 0 8 0 0 2 0 0 4 -2 -2 0 2 -1 2 -4 0 1 0 1 1 3 -3 0 -2 1 2 1 0 1 -1 1
1 -3 0 1
0 0 -1 2 -2 0 -2 1 -2 0 1 -2 1 -2 -1 -2 0 0 -1 0 -1 0 0 0 2 0 -2 -2 0 -2 2 0 -2
-2 -2 0 -2 0 0 8 0 0 -2 0 -2 -2 0 2 1 0 -2 1 -2 -1 -1 1 -3 -1 0 -2 0 -2 -1 2
0 0 -2 0 -2 -1 0 1
2 -1 1 -1 -1 1 -1 -2 0 1 -1 1 -1 -1 0 0 -2 3 1 -1 0 0 -1 -1 -4 -4 2 0 0 -4 4 0
2 -2 0 0 0 4 0 0 8 0 2 -4 2 -2 2 4 0 -3 2 -1 -1 -1 1 -2 1 0 -1 1 -1 1 0 0 2
3 2 -3 1 -1 0 1
1 2 -1 0 1 1 -1 1 -1 0 0 -1 1 -1 -1 0 -3 -2 -2 2 -1 1 1 0 4 -4 2 -4 -4 2 -2 -2
0 2 -4 0 -4 0 2 0 0 8 0 0 4 -2 -2 0 3 0 0 -2 -1 2 -2 0 -1 1 -2 -1 -1 -1 0 0
-3 2 -2 2 1 0 0 0
3 1 -1 -2 -2 1 -1 1 -1 1 -1 0 -1 -1 1 2 -2 2 -2 2 1 0 0 -2 -4 -2 4 0 4 -2 2 -4
4 -2 2 4 0 2 0 -2 2 0 8 -4 0 0 2 2 1 0 1 -2 0 0 0 -1 1 0 0 2 -1 0 1 1 -1 2 2
0 1 0 1 -1
-3 -1 0 2 1 -2 2 -1 0 -3 1 -1 1 1 -1 0 3 -2 0 -2 0 0 0 2 4 4 -2 -2 0 4 -2 4 -2
2 0 0 0 -2 0 0 -4 0 -4 8 0 0 -2 -2 -1 1 -1 1 1 0 1 1 -1 -2 1 -1 1 0 -1 0 1
-2 -2 2 0 0 -1 1
0 1 -1 -1 1 -1 1 1 -1 0 0 0 1 0 -1 2 -1 -1 -1 0 -2 0 1 1 2 -4 4 -4 -2 2 -2 -2 2
2 -2 0 -4 2 4 -2 2 4 0 0 8 -2 0 0 1 -1 1 -3 0 0 0 0 0 1 -1 0 -1 1 1 1 0 1 -1
0 2 -1 1 1
1 0 1 -2 0 0 1 0 1 2 -1 1 -1 1 3 1 1 0 0 0 1 -2 -1 1 -2 4 -2 4 0 2 -2 0 -2 -2 4
0 4 -4 -2 -2 -2 -2 0 0 -2 8 0 -4 0 2 0 0 0 1 0 0 0 1 1 1 1 -1 2 0 0 -1 0 0 0
2 -1 -1
1 1 -2 1 -2 0 0 1 -1 1 1 0 1 -1 0 0 0 0 -1 1 -1 1 -2 -2 -4 0 0 0 2 -4 2 2 2 -2
2 2 2 2 -2 0 2 -2 2 -2 0 0 8 2 -1 1 -2 1 -1 -2 1 2 0 0 2 1 2 0 -1 0 1 -1 0 0
-1 1 2 -1
0 1 0 1 0 0 0 0 1 0 1 1 1 -1 -1 -1 -1 0 1 -1 -1 1 1 -2 -2 -4 2 -2 2 -4 4 0 2 -2
-2 0 0 4 0 2 4 0 2 -2 0 -4 2 8 -1 -1 1 0 1 -2 2 0 2 -1 0 1 1 1 -1 0 1 0 2 -2
-1 -1 2 2
2 1 0 -1 0 2 -2 1 -2 1 -1 -2 -1 -1 0 0 -2 -1 -3 3 0 -1 -1 0 2 -1 0 -1 -2 0 0 -1
1 1 -1 1 -1 0 2 1 0 3 1 -1 1 0 -1 -1 8 0 0 -4 -4 4 -4 0 -2 4 -4 -2 -4 -2 4 2
-4 4 -4 4 2 -2 -4 -2
-1 2 -2 2 0 -1 1 1 -1 -1 2 0 1 0 0 0 1 -2 -1 1 -1 0 -1 -1 1 2 0 -1 1 2 -1 1 0 1
0 1 1 -2 -1 0 -3 0 0 1 -1 2 1 -1 0 8 -4 2 2 2 0 4 -2 0 4 2 4 -4 -2 0 -4 -4
-2 4 -4 4 0 -4
0 0 2 -3 2 1 1 0 2 1 -1 3 -1 0 0 1 -1 1 1 0 1 -1 2 0 0 -2 2 1 -1 0 -1 -2 0 0 0
-2 -1 1 2 -2 2 0 1 -1 1 0 -2 1 0 -4 8 -4 2 2 0 -4 4 2 -2 2 -4 2 4 -2 2 2 4
-2 4 -2 0 2
-1 -1 1 2 -1 0 -1 -2 1 0 1 0 0 0 2 -2 1 0 2 -2 1 2 -1 -1 -1 2 -3 2 1 -1 1 2 -2
-2 1 -1 2 -1 -4 1 -1 -2 -2 1 -3 0 1 0 -4 2 -4 8 0 -2 0 0 -2 -4 4 -2 4 -2 -4
-2 0 -4 0 -2 -4 4 0 -2
-2 1 -1 1 2 -2 2 0 1 -3 2 2 1 0 -1 2 1 -1 2 -1 -1 0 2 0 0 0 2 -1 2 2 0 0 1 2 -1
0 0 1 0 -2 -1 -1 0 1 0 0 -1 1 -4 2 2 0 8 0 4 0 4 -2 2 4 2 2 -2 0 0 -4 4 0 -2
0 2 2
0 2 0 -1 2 2 0 1 0 0 0 1 0 -1 0 0 -1 -1 -1 2 1 0 0 0 2 -1 1 0 -2 2 -2 -1 0 2 -1
-1 -1 -1 1 -1 -1 2 0 0 0 1 -2 -2 4 2 2 -2 0 8 -4 0 0 4 -2 0 -2 -4 2 -2 -4 2
-2 4 2 2 -4 -4
0 -1 -1 1 0 -2 2 -1 1 -2 0 1 0 1 -1 2 1 1 1 -2 -1 -1 0 0 -2 1 1 -1 2 0 2 2 1 1
0 1 1 1 0 -1 1 -2 0 1 0 0 1 2 -4 0 0 0 4 -4 8 2 4 -2 0 4 2 4 -2 2 4 -2 2 -2
-2 -2 2 4
-1 1 -2 2 0 -1 2 2 0 -1 1 0 2 1 -2 -1 2 -1 -1 1 -1 0 -1 0 1 1 0 -2 0 1 -1 2 0 2
-1 0 0 -1 1 1 -2 0 -1 1 0 0 2 0 0 4 -4 0 0 0 2 8 0 2 0 0 4 0 -2 0 0 -2 -4 4
-2 0 2 0
1 0 0 -2 1 0 1 0 2 0 -1 2 -1 1 0 2 0 1 1 0 1 -1 1 -1 -2 -1 2 1 1 0 1 0 2 1 0 0
1 1 1 -3 1 -1 1 -1 0 0 0 2 -2 -2 4 -2 4 0 4 0 8 2 -2 4 0 4 2 -2 2 0 4 -2 2
-2 2 2
1 1 0 -2 2 2 1 2 1 2 -2 1 -1 1 -1 0 -1 0 -1 3 0 -2 -1 0 1 -1 1 0 -3 0 -2 -1 0 2
0 -1 0 -1 3 -1 0 1 0 -2 1 1 0 -1 4 0 2 -4 -2 4 -2 2 2 8 -4 0 -2 0 4 -2 -2 2
-2 2 2 -2 -2 -2
-1 0 0 1 -1 -1 0 -1 0 0 2 0 1 0 2 -1 1 0 1 -1 0 1 -1 -1 -1 2 -1 1 2 0 0 1 -1 -2
2 1 1 -1 -3 0 -1 -2 0 1 -1 1 2 0 -4 4 -2 4 2 -2 0 0 -2 -4 8 2 4 -2 -2 -2 0
-4 2 0 -2 4 2 -2
1 1 -2 -1 0 -1 1 0 0 -1 1 2 0 0 0 2 -1 1 0 1 0 -1 0 -1 -2 0 3 0 2 1 1 0 2 1 0 2
0 0 0 -2 1 -1 2 -1 0 1 1 1 -2 2 2 -2 4 0 4 0 4 0 2 8 0 0 0 0 0 0 4 0 0 0 2
0
-1 1 -1 2 0 -1 1 0 1 0 1 0 2 1 0 -1 1 -1 1 -1 -1 1 -1 -1 -1 1 -1 0 1 0 0 2 -1
-1 1 0 2 -1 -2 0 -1 -1 -1 1 -1 1 2 1 -4 4 -4 4 2 -2 2 4 0 -2 4 0 8 0 -4 -2 0
-4 0 0 -4 4 2 0
0 -2 1 -1 1 -1 1 -1 1 -1 -1 0 -1 2 -1 2 1 1 1 -1 0 -1 1 1 -1 0 0 0 0 -1 1 1 1 1
0 0 1 2 1 -2 1 -1 0 0 1 -1 0 1 -2 -4 2 -2 2 -4 4 0 4 0 -2 0 0 8 0 0 4 0 2 -2
2 -4 2 4
2 -1 2 -4 0 1 0 1 1 3 -3 0 -2 1 2 1 0 1 -1 1 1 -3 0 1 0 0 0 2 -1 0 -1 -2 0 -1 2
0 0 -1 2 -1 0 0 1 -1 1 2 -1 -1 4 -2 4 -4 -2 2 -2 -2 2 4 -2 0 -4 0 8 0 0 2 0
0 4 -2 -2 0
1 0 -2 1 -2 -1 -1 1 -3 -1 0 -2 0 -2 -1 2 0 0 -2 0 -2 -1 0 1 0 0 1 -2 2 0 2 -1 2
0 -1 2 -1 2 1 2 0 0 1 0 1 0 0 0 2 0 -2 -2 0 -2 2 0 -2 -2 -2 0 -2 0 0 8 0 0
-2 0 -2 -2 0 2
0 -3 2 -1 -1 -1 1 -2 1 0 -1 1 -1 1 0 0 2 3 2 -3 1 -1 0 1 -2 1 -1 1 1 -1 1 2 0
-1 1 -1 1 1 0 0 2 -3 -1 1 0 0 1 1 -4 -4 2 0 0 -4 4 0 2 -2 0 0 0 4 0 0 8 0 2
-4 2 -2 2 4
3 0 0 -2 -1 2 -2 0 -1 1 -2 -1 -1 -1 0 0 -3 2 -2 2 1 0 0 0 -1 -2 1 0 -1 -1 1 -1
1 0 0 1 -1 1 1 0 3 2 2 -2 1 -1 -1 0 4 -4 2 -4 -4 2 -2 -2 0 2 -4 0 -4 0 2 0 0
8 0 0 4 -2 -2 0
1 0 1 -2 0 0 0 -1 1 0 0 2 -1 0 1 1 -1 2 2 0 1 0 1 -1 -3 -1 1 2 2 -1 1 -1 1 -1 1
0 1 1 -1 -2 2 -2 2 -2 -1 0 0 2 -4 -2 4 0 4 -2 2 -4 4 -2 2 4 0 2 0 -2 2 0 8
-4 0 0 2 2
-1 1 -1 1 1 0 1 1 -1 -2 1 -1 1 0 -1 0 1 -2 -2 2 0 0 -1 1 3 1 0 -2 -1 2 -2 1 0 3
-1 1 -1 -1 1 0 -3 2 0 2 0 0 0 -2 4 4 -2 -2 0 4 -2 4 -2 2 0 0 0 -2 0 0 -4 0
-4 8 0 0 -2 -2
1 -1 1 -3 0 0 0 0 0 1 -1 0 -1 1 1 1 0 1 -1 0 2 -1 1 1 0 -1 1 1 -1 1 -1 -1 1 0 0
0 -1 0 1 -2 1 1 1 0 2 0 -1 -1 2 -4 4 -4 -2 2 -2 -2 2 2 -2 0 -4 2 4 -2 2 4 0
0 8 -2 0 0
0 2 0 0 0 1 0 0 0 1 1 1 1 -1 2 0 0 -1 0 0 0 2 -1 -1 -1 0 -1 2 0 0 -1 0 -1 -2 1
-1 1 -1 -3 -1 -1 0 0 0 -1 2 1 -1 -2 4 -2 4 0 2 -2 0 -2 -2 4 0 4 -4 -2 -2 -2
-2 0 0 -2 8 0 -4
-1 1 -2 1 -1 -2 1 2 0 0 2 1 2 0 -1 0 1 -1 0 0 -1 1 2 -1 -1 -1 2 -1 2 0 0 -1 1
-1 -1 0 -1 1 0 0 0 0 1 -1 1 -1 2 2 -4 0 0 0 2 -4 2 2 2 -2 2 2 2 2 -2 0 2 -2
2 -2 0 0 8 2
-1 -1 1 0 1 -2 2 0 2 -1 0 1 1 1 -1 0 1 0 2 -2 -1 -1 2 2 0 -1 0 -1 0 0 0 0 -1 0
-1 -1 -1 1 1 1 1 0 -1 1 1 -1 -1 2 -2 -4 2 -2 2 -4 4 0 2 -2 -2 0 0 4 0 2 4 0
2 -2 0 -4 2 8
-
DET
1
-
MINIMAL_NORM
8
-
KISSING_NUMBER
6218175600
-
SUBGROUP_ORDER
2^8*3^2*5^2*7*13
-
SUBGROUP
SL2(25)xPSL2(7):2
-
SUBGROUP_GENERATORS
#6
72
0 -1 -1 -1 -1 2 2 0 0 0 1 -1 0 0 0 0 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0
1 0 1 2 5 -6 -2 -1 0 2 -2 1 1 -2 0 0 2 2 -2 2 1 1 -1 -1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0
0 -2 0 0 -1 3 2 1 -2 -1 0 0 1 0 1 0 -1 -1 1 -1 0 -1 1 -1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0
1 2 1 1 3 -5 -1 -2 1 2 -1 -1 0 -2 -1 0 2 2 -1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0
1 -1 1 2 3 -2 -2 1 -2 0 -2 2 1 0 1 0 1 1 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0
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1 -1 0 -1 0 0
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1 1 -1 -1
0 3 1 2 1 -2 -1 -1 0 1 -2 0 0 -1 0 0 2 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 3 -2 0 0 -1 1 -1 0 0 -1 1 -1 1 1 -1
0 0 1 0 0
-2 1 -2 -1 -6 7 2 -1 0 -2 1 0 0 3 1 0 -2 -2 2 0 -1 -2 1 2 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 0 -1 -1 1 2 -1 1 -1 -1 0 -1 0 0 0
-1 -1 0 0 -1 0
2 1 0 1 4 -4 -2 -1 -1 1 -2 1 -1 0 0 -1 2 1 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 -3 4 1 0 0 0 2 0 0 1 0 1 -1 -1 1 0
-1 -1 1 1
1 1 -1 1 -1 3 1 -1 0 0 1 0 -1 1 0 0 1 0 1 0 -1 0 1 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 -2 3 1 0 -2 -2 -1 1 2 1 2 0 -1 -1 1
0 0 -2 1 -1
0 0 0 0 -2 2 0 -1 1 -1 0 1 1 0 0 0 1 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 1 -1 1 -1 -1 0 -1 0 1 -1 0 -1 0 -1
0 0 1 0 0 -1
-
PROPERTIES
Unimodular=1, Extremal=1
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REFERENCES
G. Nebe: An even unimodular 72-dimensional lattice of minimum 8.
preprint 2010</A>
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NOTES
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe