The Lattice 2^1+10.O10^+(2) = BW32
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 13:14:03 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM (floating point or integer Gram matrix)
DIVISORS (elementary divisors)
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_GENERATORS
PROPERTIES
REFERENCES
NOTES
URL (links to other sites for this lattice)
LAST_LINE
-
NAME
2^1+10.O10^+(2) = BW32
-
DIMENSION
32
-
GRAM (floating point or integer Gram matrix)
32 0
4
1 4
0 1 4
0 0 0 4
-1 0 0 0 4
0 0 0 -1 0 4
0 0 0 0 -2 -1 4
0 -1 1 -1 0 0 0 4
1 1 0 -1 0 2 -1 -1 4
0 0 0 1 0 0 0 0 -1 4
-1 -1 -1 0 0 0 0 -1 0 0 4
0 1 -1 0 1 -1 0 -1 -1 1 0 4
-1 -1 -1 0 1 1 -1 0 0 1 -1 0 4
0 -1 0 0 0 0 0 0 1 -1 1 -2 0 4
0 1 0 0 -1 0 1 -1 0 0 -1 0 0 1 4
0 -1 -1 1 -1 -1 0 -1 -1 0 1 1 0 -1 -1 4
0 0 -1 0 1 0 0 0 0 0 1 1 -1 0 -1 0 4
0 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 4
0 0 1 0 0 0 0 0 0 -1 -1 -1 1 0 1 -1 -2 -1 4
0 -1 0 -1 0 0 0 1 0 -1 1 -1 -1 0 -1 0 1 1 -1 4
-1 0 1 1 1 0 -1 -1 0 1 1 0 0 0 -1 0 0 0 0 -1 4
0 0 -1 1 0 0 0 -1 0 1 1 0 1 1 1 0 0 1 -1 0 0 4
-1 0 0 0 0 1 1 0 1 -1 -1 0 1 0 1 -1 0 0 1 0 -1 0 4
0 0 0 0 1 -1 0 -1 -1 -1 1 0 -1 0 -1 1 1 0 0 1 1 -1 -2 4
0 0 0 -1 1 0 -1 0 0 -1 0 0 1 0 -1 0 -1 1 1 0 1 0 0 1 4
0 -1 0 0 0 -1 0 0 -1 0 0 0 1 0 0 0 -2 0 2 0 0 0 0 0 2 4
-1 -2 0 1 0 -1 0 0 -1 0 2 -1 0 1 -1 1 0 0 0 1 2 1 -1 1 0 0 4
0 -1 0 0 1 0 -1 0 0 1 0 0 1 1 0 -1 -1 -1 1 -1 1 -1 -1 0 0 2 0 4
-1 0 1 1 1 0 -1 0 0 0 -1 0 1 -1 0 0 -1 -1 2 -1 2 -1 1 0 1 0 1 0 4
0 -2 -1 -1 0 1 0 1 1 -2 1 -1 0 1 -2 0 1 -1 0 1 0 -1 1 0 1 0 1 0 0 4
0 0 1 -1 1 1 -1 1 1 -2 -1 -1 0 0 -1 -1 0 0 1 1 1 -2 1 1 2 0 0 0 2 2 4
1 2 0 0 -1 0 1 0 0 1 -1 1 -1 -1 1 0 1 1 -2 0 -2 1 0 -1 -2 -2 -2 -2 -2 -2 -2 4
-
DIVISORS (elementary divisors)
1
-
MINIMAL_NORM
4
-
KISSING_NUMBER
146880
-
GROUP_ORDER
2^31 * 3^5 * 5^2 * 7 * 17 * 31
-
GROUP_GENERATORS
3
32
0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 -1 -2 1 1 0 0 -1 1 1 -1 -2 1 -2
0 0 0 0 0 0 0 0 0 -1 1 0 1 1 0 -1 0 1 -1 0 0 -1 -1 1 -1 3 0 0 4 2 -1 4
0 0 0 0 0 -1 -1 0 0 1 0 0 0 0 0 0 -1 0 0 -1 0 0 1 1 -1 0 0 -1 -1 1 1 0
-1 0 0 1 1 -1 1 0 1 2 -1 1 -1 -1 0 2 -1 -2 4 1 0 0 1 -3 3 -6 2 0 -9 -4 4 -4
0 0 1 1 0 -1 -1 1 1 0 1 0 2 -1 3 -1 -1 0 -1 1 2 -1 0 3 0 2 -1 0 2 6 -3 5
1 0 -1 -1 0 1 0 0 -1 -2 1 -1 0 1 -1 0 0 0 -1 0 1 0 1 1 0 2 0 2 4 0 -1 5
-1 1 0 0 0 0 0 0 0 1 -1 0 -1 0 0 1 0 0 1 0 -1 1 -1 -1 0 -1 0 0 -1 0 1 -2
0 0 0 0 1 0 0 -1 0 1 -1 0 0 0 0 0 0 0 -1 -1 -1 0 -1 -1 -2 1 0 -3 -2 -1 2 -4
1 0 0 -1 -1 1 -1 1 0 -2 1 -1 0 0 0 -1 0 1 -3 -1 0 0 2 4 0 4 1 3 7 3 -4 7
0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 -1 0 0 -2
0 0 0 -1 0 1 1 -1 -1 0 0 0 -1 1 -2 1 1 -1 1 0 -1 1 0 -2 1 -1 1 1 0 -3 2 0
0 0 1 1 0 -1 -1 1 1 0 1 0 2 -1 3 -1 0 1 0 1 1 -1 -1 2 0 1 -1 0 2 5 -3 3
0 0 0 1 0 0 0 1 1 -1 0 0 1 -1 2 0 0 0 0 1 2 -1 0 1 1 0 0 1 0 2 -2 2
-1 0 0 0 0 0 1 0 0 1 -1 0 -2 -1 -1 2 0 -1 2 0 -1 2 1 -1 3 -4 1 3 -3 -2 2 -1
0 1 0 0 0 0 0 1 0 0 0 -1 0 0 1 0 0 1 -1 0 0 0 0 2 0 1 -1 1 3 3 -3 1
0 0 0 0 0 0 0 0 0 1 -1 1 -1 0 -1 1 0 -1 2 1 0 0 1 -2 1 -4 0 -1 -5 -4 1 -6
0 0 0 0 1 -1 0 0 0 1 0 0 0 -1 0 1 -1 -2 2 1 1 0 1 -2 2 -3 1 0 -5 -2 2 -1
-1 -1 -1 -1 0 1 2 -2 -1 -1 0 0 -1 2 -3 1 2 1 2 0 -1 1 -3 -3 -1 1 -1 1 3 -2 3 2
0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 1 0 1 1 0 1 0 2
1 0 0 -2 0 1 0 -1 -1 -1 0 -1 0 1 -2 -1 1 0 -3 -2 -1 0 0 0 -3 4 0 -2 3 -2 0 -1
0 -1 0 1 0 -1 1 0 0 0 1 1 1 0 0 1 -1 -1 3 2 2 0 0 -2 2 -4 -1 1 -3 -1 2 1
-1 0 0 0 0 0 1 0 1 0 0 -1 -1 -1 0 1 1 0 1 0 -1 1 0 0 2 -1 1 3 1 0 0 2
0 0 0 0 0 0 -1 1 1 0 0 -1 0 -1 2 0 0 0 -1 -1 0 -1 1 2 1 2 2 1 1 2 -1 4
0 0 0 0 0 0 1 -1 -1 0 0 1 0 1 -2 0 0 -1 1 1 0 0 -1 -3 0 -2 -1 -1 -2 -3 2 -3
-1 -2 -1 1 0 1 3 -2 0 -1 0 1 -1 1 -3 1 3 0 3 0 -2 0 -3 -6 2 -2 1 1 -2 -7 6 -1
-1 -1 -1 1 0 1 2 -1 0 0 -1 1 -1 1 -2 1 2 0 3 0 -1 0 -2 -4 1 -2 1 0 -3 -5 4 -2
0 0 1 0 0 -1 0 0 0 1 0 0 0 -1 1 1 -1 -1 1 1 1 1 1 0 1 -3 -1 1 -2 1 0 -1
0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 -1 0 1 1 2 0 0 1 0 -1 -1 0 -1 2 -1 1
1 0 1 1 0 -1 -1 1 1 0 1 0 2 -1 3 -1 -1 0 -2 0 1 -2 1 2 0 1 0 -1 0 3 -2 1
0 -1 0 0 0 0 0 0 0 0 0 0 -1 -1 0 1 0 -1 1 0 0 1 1 -1 2 -2 1 2 -2 -2 2 1
1 -1 0 0 0 0 0 0 0 -1 1 0 1 0 0 -1 0 0 -2 -1 0 -1 0 0 -1 2 0 -1 1 -1 1 1
0 1 0 -1 0 0 -1 0 0 0 0 -1 0 0 0 -1 0 1 -2 -1 -1 0 0 2 -2 3 0 -1 3 2 -2 0
32
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 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 0 -1 -1 0 -1
0 0 0 0 1 0 0 0 0 0 0 -1 0 -1 1 0 0 0 -1 -1 0 0 0 0 0 1 1 0 0 1 0 1
0 0 0 1 1 -1 0 0 0 1 1 0 1 -1 2 0 -1 0 1 1 1 -1 -1 -1 0 -1 -1 -1 -2 2 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 -1 -1 0 0 1 0 -1 0 1 0 1
0 0 0 0 -2 0 -1 1 0 -1 -1 0 0 0 0 -1 0 1 -1 1 1 1 1 3 0 0 -2 2 4 3 -5 1
0 1 1 0 0 -1 -1 1 0 1 0 0 0 -1 1 0 -2 -1 0 1 1 1 2 2 0 -2 -1 0 -2 2 -2 -2
0 -1 -1 0 1 1 2 -1 0 -1 0 0 0 1 -2 0 1 0 0 -1 -1 -1 -2 -3 -1 1 1 -2 -1 -5 3 -2
0 0 0 0 -1 0 -1 1 1 1 -1 0 -1 -1 0 0 0 0 0 -1 -1 1 2 2 1 -1 1 1 -1 0 -1 -1
0 0 -1 0 0 1 1 0 -1 -1 0 1 0 2 -1 0 0 1 1 2 2 0 -2 -1 -1 1 -2 0 3 1 -1 2
0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 -1 2 1 0 0 0 -2 2 -3 1 1 -3 -2 2 -1
0 1 0 0 0 0 -1 1 0 0 0 1 1 1 1 0 -1 0 0 1 2 -1 0 1 -1 1 -1 -2 0 2 -2 0
0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 -1 0 1 -1 0 0 0 0 2 -1 2 -1 0 3 2 -2 1
0 0 0 0 0 0 1 -1 0 1 0 0 0 0 -1 0 1 0 0 -1 -2 1 -1 -1 -1 0 0 -1 -1 -2 3 -3
-1 0 0 0 0 0 0 -1 0 2 -1 1 -1 1 -1 0 0 0 1 0 -1 1 -1 -1 -1 -1 -1 -2 -2 -1 2 -4
0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 1 0 3
0 0 1 1 -1 -1 -1 2 1 -1 1 0 2 -1 3 -1 -1 1 -1 1 2 0 0 4 0 1 -2 1 3 6 -4 4
0 1 0 -2 -1 2 0 -1 -1 -1 0 -1 -1 1 -2 -1 2 1 -4 -3 -4 1 0 1 -2 5 2 1 6 -1 0 1
0 0 1 1 1 -1 -1 0 1 2 -1 0 0 -2 2 0 -1 -1 0 0 0 -1 1 0 1 -2 1 -2 -5 0 0 -4
-1 -1 0 0 0 0 1 -1 0 0 0 0 -1 0 -1 1 1 0 2 -1 -2 1 -1 -2 1 -1 2 2 -1 -2 4 2
2 2 1 -1 -1 0 -3 2 0 -1 1 -2 2 -1 4 -3 -1 2 -7 -1 1 -1 2 8 -3 7 -1 0 10 10 -9 5
0 0 0 0 -1 1 0 -1 -1 0 1 0 0 1 -1 -1 1 1 -1 -1 -2 0 -1 0 -1 2 0 0 3 0 1 1
0 1 1 0 -1 -1 -3 2 1 2 -1 0 0 -2 2 -1 -2 0 -2 -1 0 1 3 5 -2 0 -1 -2 -2 3 -3 -5
0 0 1 0 0 -1 0 0 0 -1 1 -1 1 -1 2 0 0 0 0 1 1 0 0 1 2 0 0 3 3 4 -2 6
1 1 1 -1 -1 0 -2 1 0 -1 1 -2 1 -1 2 -1 0 1 -4 -1 0 0 2 5 -1 4 0 2 7 5 -5 5
0 0 0 0 1 0 0 -1 0 1 0 0 0 0 0 1 0 -1 1 0 0 -1 0 -2 0 0 2 -1 -3 -2 3 0
1 1 1 0 0 0 -1 0 0 0 1 -1 1 -1 2 -2 0 1 -4 -2 -2 -1 0 3 -2 4 1 -1 4 4 -2 1
0 0 -1 0 1 0 1 -1 0 1 -1 1 0 1 -1 1 0 -1 2 1 1 -1 -1 -3 0 -1 1 -2 -4 -3 3 -2
1 1 1 0 0 -1 -3 2 1 1 0 -1 1 -2 4 -2 -2 1 -4 -1 1 -1 2 6 -2 3 -1 -2 3 7 -6 0
1 1 1 0 -1 -1 -2 2 1 0 0 -1 1 -2 2 -1 -1 0 -3 -1 0 0 3 5 -1 1 0 0 1 3 -4 -1
1 1 1 -1 -1 -1 -3 2 1 0 0 -2 1 -2 3 -2 -1 1 -5 -2 0 0 3 7 -2 4 0 0 5 6 -6 2
-1 -1 -1 0 0 1 2 -1 -1 -1 0 1 -1 2 -3 1 1 0 3 1 0 1 -2 -4 1 -2 -1 1 -1 -4 3 0
32
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 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
-1 -1 0 1 0 -1 0 0 1 1 0 0 -1 -1 0 1 0 -1 3 0 0 0 1 -1 3 -3 2 2 -4 -2 3 2
1 1 1 0 0 -1 -2 2 1 1 0 0 1 -1 2 -1 -2 0 -3 -1 1 0 2 4 -3 2 -1 -3 0 3 -3 -3
0 0 0 0 -1 0 -1 1 0 -1 0 0 0 0 0 -1 -1 1 -1 1 1 1 1 3 0 0 -2 2 4 3 -4 2
0 0 0 0 0 -1 -1 1 0 1 0 0 0 -1 1 1 -1 -1 2 1 2 1 2 1 1 -3 -1 1 -3 1 0 0
0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 -1 1 1 -1 -2 -2 -1 -1 -1 -2 2 1 -3 -1 -1 2 -3
0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 1 1 0 1 2 -1 -1 0 1 1
0 0 0 0 0 0 1 -1 -1 -1 0 0 0 1 -1 1 1 0 2 2 1 1 -1 -2 1 -2 -2 2 1 0 0 1
0 0 0 0 0 0 0 0 0 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 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 -1 1 -1 0 -1 1 0 0 1 -1 0 0 0 0 1 1 1 0 3 4 3 -2 5
0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 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 -1 0 0 1 1 0 0 0 0 -1 -1 0 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 1 1 1 0 1 1 0 0 0 2
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 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 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 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 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 1 0 0 0 0 -1 0 -1 0 -1 -1 0 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 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 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 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 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 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 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 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 1
-
PROPERTIES
INTEGRAL = 1
MODULAR = 1
-
REFERENCES
G. Nebe, Finite quaternionic matrix groups,
Representation Theory 2, 106-223 (April 1998)
-
NOTES
-
URL (links to other sites for this lattice)
All the quaternionic matrix groups
-
LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe