The Lattice 4.L3(4).2^2a
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 13:14:24 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
4.L3(4).2^2a
-
DIMENSION
32
-
GRAM (floating point or integer Gram matrix)
32 0
8
4 8
3 4 8
4 4 2 8
4 4 4 4 8
3 3 1 3 3 8
4 4 2 3 3 3 8
4 2 4 4 4 3 4 8
3 4 4 4 4 3 3 4 8
4 3 2 4 4 2 4 4 4 8
3 3 3 1 2 3 3 1 0 1 8
3 4 2 3 2 4 4 2 3 4 2 8
4 2 4 3 4 1 4 4 3 3 1 3 8
3 2 2 1 2 3 4 3 1 2 2 4 2 8
4 4 2 4 3 3 4 4 2 4 3 2 2 1 8
3 3 3 1 2 1 2 2 4 2 0 1 2 2 1 8
3 1 1 1 1 3 1 1 1 1 3 1 1 1 3 1 8
4 2 2 3 4 2 4 3 2 4 1 4 4 2 2 2 2 8
3 2 2 2 2 4 2 2 2 2 1 4 2 3 4 1 3 2 8
4 3 3 2 3 1 2 2 1 1 3 1 4 0 4 1 4 2 2 8
3 4 4 3 3 3 4 4 4 4 3 4 3 2 4 2 3 4 2 3 8
3 2 1 0 2 2 0 0 2 2 1 2 0 2 1 4 1 2 1 1 2 8
4 2 2 4 3 2 2 3 4 4 -1 3 4 2 2 4 1 4 2 2 2 4 8
4 2 4 2 2 1 2 4 2 2 2 1 2 3 2 2 2 2 1 2 2 1 2 8
4 3 2 2 3 2 2 2 4 4 1 2 2 2 2 4 1 2 2 0 2 4 4 2 8
4 4 4 2 2 1 2 2 3 2 2 3 4 1 3 1 2 2 2 4 2 1 3 4 2 8
3 1 3 3 4 3 1 4 4 2 0 1 3 1 1 4 3 4 1 1 3 2 4 2 2 1 8
3 1 4 3 3 3 3 4 3 1 3 1 4 1 2 1 1 3 2 2 3 -1 2 2 1 2 3 8
3 2 3 3 4 3 3 4 4 3 1 3 4 3 3 2 3 4 3 2 4 2 4 2 2 2 4 4 8
3 2 3 1 1 4 2 1 1 2 4 2 1 2 2 1 3 1 2 1 3 1 1 2 2 2 2 3 1 8
3 4 3 3 3 3 3 1 3 2 3 4 3 1 2 1 1 3 2 1 3 2 1 1 2 4 2 2 1 3 8
2 2 4 4 2 2 2 4 2 2 2 2 3 2 2 1 2 2 2 2 2 -2 2 3 0 2 3 2 2 2 1 8
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DIVISORS (elementary divisors)
1^16 5^16
-
MINIMAL_NORM
8
-
KISSING_NUMBER
21600
-
GROUP_ORDER
2^10 * 3^2 * 5 * 7
-
GROUP_GENERATORS
3
32
1 2 -1 -4 -1 -1 -4 0 3 0 0 1 0 2 3 -1 0 1 -2 1 -1 -3 3 1 0 -4 -1 1 0 0 3 0
1 3 -2 -5 -1 -2 -5 -1 4 0 -1 2 1 3 5 -2 1 1 -3 1 -2 -2 2 1 0 -5 0 3 -1 0 3 1
2 -2 1 -2 1 1 0 -2 -2 1 -2 2 -3 0 1 1 0 -2 -2 1 1 -2 1 0 1 0 0 2 1 -2 2 1
1 2 -1 -4 0 -1 -4 -1 3 -1 -1 2 1 2 4 -1 1 1 -3 0 -1 -2 2 1 0 -4 -1 2 -1 0 2 1
3 1 -1 -7 -1 0 -5 -3 2 1 -2 2 -2 3 5 -1 -1 -1 -4 3 0 -5 4 2 1 -6 0 3 1 -2 6 2
1 1 -1 -3 0 -1 -1 -2 0 1 -2 2 -1 1 2 -1 0 -2 -2 1 0 -1 1 1 1 -2 1 3 0 -1 2 2
3 1 0 -6 0 0 -2 -3 -1 2 -3 2 -3 2 3 -1 0 -3 -3 2 1 -4 4 1 1 -4 1 4 0 -3 5 2
2 -2 1 -2 1 1 0 -2 -2 1 -1 1 -3 0 0 1 -1 -2 -1 1 2 -3 2 0 1 0 0 1 1 -2 2 1
2 2 -1 -7 -1 -1 -4 -3 2 1 -2 3 -2 3 5 -2 0 -1 -4 3 -1 -4 4 2 1 -6 1 4 0 -2 5 2
1 3 -2 -6 -1 -2 -5 -1 4 0 -1 2 0 3 5 -2 0 1 -3 2 -2 -3 3 2 0 -6 0 3 -1 0 4 1
2 1 -1 -5 1 -1 -1 -2 -1 1 -4 3 -2 1 3 -1 1 -3 -3 1 0 -1 2 1 1 -3 0 5 0 -2 4 3
1 3 -2 -6 0 -2 -3 -2 1 1 -3 3 -1 2 4 -2 1 -2 -3 2 -1 -2 3 2 1 -5 1 5 -1 -1 4 2
2 -2 2 -1 0 2 0 -2 -2 1 -1 0 -3 0 0 1 -1 -1 -1 1 2 -3 2 0 1 0 0 0 1 -2 2 0
1 -1 0 -1 1 0 2 -1 -4 2 -2 1 -3 -1 -2 1 0 -4 0 1 2 -1 1 0 1 1 1 2 1 -2 2 1
1 0 0 -1 0 0 -2 0 1 0 0 0 0 1 1 0 0 1 -1 0 0 -2 1 0 0 -1 -1 0 0 0 1 0
0 0 -1 -1 0 -1 -1 0 1 0 0 1 -1 1 1 0 0 0 -1 1 0 -1 1 0 0 -1 0 1 0 0 1 0
0 -2 1 3 0 1 2 1 -2 0 1 -1 -1 -1 -3 1 -1 0 1 0 1 0 -1 -1 1 3 0 -2 1 0 -1 -1
2 -2 1 -1 1 1 0 -1 -2 0 -1 1 -3 0 0 1 -1 -2 -1 1 2 -3 2 0 1 0 0 0 1 -1 2 0
1 -2 1 1 1 1 2 -1 -3 1 -1 0 -2 -1 -2 1 0 -2 0 0 2 -1 0 0 1 2 0 0 1 -1 0 0
1 0 0 -1 0 0 -2 0 1 0 0 0 0 1 1 0 0 1 -1 0 0 -1 1 0 0 -1 -1 0 0 0 1 0
1 -3 2 1 1 1 1 -1 -2 0 -1 1 -2 0 0 1 0 -1 -1 0 1 -1 0 -1 1 2 0 0 0 -1 0 0
-1 2 -2 -2 -1 -2 -3 1 3 0 0 1 0 2 2 -1 0 1 -1 2 -2 -1 1 1 0 -3 0 1 0 1 2 0
0 1 -1 -2 0 -1 -2 0 1 0 0 1 -1 1 1 0 0 0 -1 1 0 -2 2 1 0 -2 0 1 0 0 2 0
0 1 -1 -1 0 -1 -1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 -1 0 -1 0 1 0 1 -1 -1 1 0 0 -1 1 0 1 0 -2 1 0 0 0 0 -1 1 0 1 -1
1 2 -1 -3 -1 -1 -3 0 2 0 0 1 0 1 2 -1 0 1 -1 1 -1 -2 2 1 0 -3 0 1 0 0 2 0
1 -2 1 0 0 1 0 -1 -1 0 0 1 -2 0 0 1 -1 -1 -1 1 1 -2 1 0 1 0 0 0 1 -1 1 0
3 -3 2 -2 2 2 2 -3 -4 1 -3 2 -4 -1 0 1 0 -4 -2 0 3 -2 2 0 1 1 0 2 1 -3 2 2
2 -3 2 -1 1 2 2 -2 -5 2 -2 1 -5 -1 -2 1 -1 -4 -1 2 3 -3 2 0 2 1 1 1 2 -3 3 1
0 -2 2 3 0 1 2 0 -1 0 0 0 0 -1 -1 1 0 0 0 -1 0 1 -2 -1 0 3 0 -1 0 0 -2 -1
2 1 0 -4 -1 0 -3 -2 2 0 -2 2 -1 2 4 -1 0 0 -3 2 -1 -3 2 1 1 -4 0 2 0 -1 3 1
1 -3 2 2 2 1 3 -1 -4 0 -1 1 -2 -2 -2 2 0 -2 0 -1 2 0 -1 -1 1 3 0 0 1 -1 -1 0
32
-2 -3 3 8 -1 2 2 2 2 -2 4 -4 3 -1 -3 1 -1 5 3 -2 -1 2 -4 -2 -1 5 -1 -7 0 3 -6 -4
-3 -1 2 7 -2 1 1 3 2 -1 4 -4 3 0 -3 1 -1 5 3 -1 -2 2 -4 -2 -1 4 0 -6 0 3 -5 -4
-2 -2 3 7 -2 2 2 2 1 -1 4 -4 3 -1 -3 1 -1 4 3 -2 -1 2 -3 -2 -1 4 0 -6 0 2 -5 -4
-1 3 -1 -2 -1 -1 -3 0 4 0 0 0 3 2 3 -2 1 2 -1 0 -3 1 0 1 -1 -3 0 1 -2 1 0 0
-1 1 1 1 -2 0 -1 1 3 0 2 -2 2 1 0 -1 0 3 1 0 -2 0 0 0 -1 -1 0 -2 -1 1 -1 -2
-1 -4 3 6 1 2 4 1 -3 0 2 -3 -1 -2 -5 2 -1 0 3 -1 2 1 -2 -2 0 6 0 -4 1 0 -3 -2
-1 -3 3 6 -1 2 3 1 -1 0 3 -3 0 -1 -4 1 -2 2 3 0 0 1 -3 -2 0 5 1 -5 1 1 -4 -3
-2 -1 2 6 -1 1 2 2 1 -1 3 -3 3 -1 -3 0 0 3 3 -2 -1 3 -3 -2 -1 4 0 -4 -1 2 -5 -3
-2 -2 2 6 -1 1 2 2 1 -1 3 -3 2 -1 -3 1 -1 3 3 -1 -1 2 -3 -2 -1 4 0 -5 0 2 -4 -3
-1 2 0 0 -2 -1 -3 1 5 -1 2 -1 3 2 2 -2 0 4 0 0 -3 0 0 0 -1 -2 0 -2 -1 2 -1 -2
-1 -4 4 7 -1 3 4 1 -2 0 3 -4 0 -2 -5 2 -2 2 3 -1 1 1 -3 -2 0 6 0 -6 2 1 -4 -3
-2 -2 2 6 0 1 2 2 0 -1 3 -3 2 -1 -3 1 -1 3 3 -2 0 2 -3 -2 -1 5 0 -5 0 2 -5 -3
-1 1 0 1 -2 0 -2 1 4 -1 3 -2 3 1 1 -1 -1 4 1 0 -2 0 0 0 -1 -1 0 -3 -1 2 -2 -2
-2 -4 3 9 0 2 4 2 -1 -1 4 -4 2 -2 -5 2 -1 3 4 -3 1 3 -5 -3 -1 8 0 -6 0 2 -7 -4
-1 1 0 0 -1 0 -1 1 2 0 1 -1 1 1 0 -1 0 2 0 1 -2 0 0 0 0 -1 0 -1 0 1 0 -1
-2 -2 2 6 -2 1 0 3 3 -2 4 -3 3 0 -1 1 -1 5 2 -1 -2 1 -3 -2 -1 3 -1 -6 0 3 -4 -4
0 -3 2 3 0 2 2 0 -1 0 1 -2 -1 -1 -2 1 -1 1 1 0 1 0 -1 -1 0 3 0 -3 1 0 -1 -1
0 -2 3 3 -1 2 1 0 0 0 1 -2 0 0 -1 0 -1 2 1 0 0 0 -1 -1 0 2 0 -3 0 0 -2 -2
0 -1 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
-1 -2 2 4 -1 2 2 1 -1 0 2 -3 0 -1 -3 1 -1 2 2 0 0 0 -1 -1 0 3 0 -4 1 1 -2 -2
-2 -3 4 7 -2 3 3 1 -1 0 3 -4 1 -1 -4 1 -2 3 3 0 0 1 -3 -2 0 5 1 -6 1 1 -4 -4
-1 -4 3 6 0 2 3 1 -1 -1 2 -3 0 -1 -3 2 -1 2 2 -1 1 0 -2 -2 0 5 -1 -5 1 1 -3 -3
0 1 0 -1 -1 0 -3 0 4 -1 1 -1 2 2 3 -1 0 3 -1 0 -2 -1 1 0 -1 -2 -1 -1 -1 1 0 -1
-2 1 0 4 -2 0 0 2 3 -1 3 -3 4 0 -1 -1 0 4 2 -2 -2 3 -3 -1 -1 2 0 -3 -1 2 -4 -2
-1 -1 1 3 -1 0 -1 1 4 -2 2 -1 3 1 1 0 0 4 0 -1 -2 1 -2 -1 -1 1 -1 -3 -1 2 -3 -2
-2 -1 1 6 -1 1 1 3 2 -2 4 -4 3 -1 -3 1 -1 5 3 -2 -1 2 -3 -2 -1 4 -1 -6 0 3 -5 -3
-1 -1 2 3 -1 1 0 1 2 -1 2 -2 2 0 0 0 0 3 1 -1 -1 1 -1 -1 -1 1 -1 -3 -1 1 -2 -2
0 -3 3 4 -1 3 2 0 -1 0 2 -3 0 -1 -3 1 -2 1 2 0 1 0 -1 -1 0 3 0 -4 1 0 -2 -2
-1 -1 1 3 -1 1 1 1 1 0 2 -3 1 0 -2 0 -1 2 2 0 0 1 -1 -1 -1 2 0 -3 0 1 -2 -2
0 -3 3 4 0 2 2 0 -1 -1 2 -2 0 -1 -2 1 -1 1 1 -1 1 0 -1 -1 0 3 0 -4 1 0 -2 -2
-1 -2 2 4 0 1 2 1 0 -1 2 -2 1 -1 -2 1 -1 2 2 -1 0 1 -2 -1 0 3 0 -4 0 1 -3 -2
-1 0 1 3 0 0 1 1 0 0 1 -1 2 -1 -1 0 1 1 1 -2 -1 3 -2 -1 -1 2 0 -1 -1 1 -3 -1
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
0 1 0 0 0 0 0 1 0 0 1 -1 0 0 -1 0 0 0 1 0 0 -1 1 0 0 0 0 -1 0 0 0 -1
-1 -3 2 7 0 2 4 1 -2 0 2 -3 0 -2 -5 2 -1 1 3 -1 1 2 -4 -2 0 6 0 -4 1 1 -4 -2
0 -3 3 6 0 2 4 1 -3 0 2 -3 -1 -2 -5 2 -1 0 3 -1 2 1 -2 -2 0 5 0 -4 1 0 -3 -2
0 2 -1 0 0 -1 0 0 0 0 -1 0 1 0 0 -1 1 0 0 -1 0 2 -1 0 0 0 0 2 -1 0 -1 1
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 1 0 0 0 0 -1 0
0 0 0 1 0 0 1 0 0 0 0 -1 0 0 -1 0 0 0 1 0 0 1 -1 0 0 1 0 0 0 0 -1 0
0 -1 1 3 0 1 2 1 -2 0 1 -2 -1 -1 -3 1 -1 0 2 0 1 0 -1 -1 0 3 0 -2 1 0 -1 -1
0 -1 0 2 1 0 2 0 -2 0 0 0 -1 -1 -2 1 0 -1 1 0 1 1 -1 0 0 2 0 0 0 0 -1 0
0 3 -2 -3 0 -2 -2 0 2 0 0 1 1 1 2 -2 1 0 -1 0 -1 0 2 1 0 -3 0 2 -1 0 1 1
0 1 -1 0 1 -1 1 0 -1 0 -1 1 0 -1 0 0 1 -1 0 -1 0 2 -1 0 0 1 0 2 -1 0 -1 1
1 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 2 -1 -2 0 -1 -1 0 1 0 -1 1 1 0 2 -1 1 0 -1 0 -1 1 0 1 0 -2 0 2 -1 0 0 1
0 0 0 1 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 0 0 1 0 0 0 0 -1 0
1 2 -1 -3 0 -1 -2 0 1 0 -1 1 0 1 2 -1 0 0 -1 1 -1 -1 1 1 0 -3 0 1 0 0 2 1
1 2 -1 -3 0 -1 -2 0 1 0 -1 1 0 1 2 -1 1 0 -2 0 -1 -1 2 1 0 -3 -1 2 0 0 2 1
1 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 1 -1 0 1 -1 1 1 -1 0 -1 1 0 -1 -1 0 1 -1 0 -1 0 2 -1 0 0 1 0 2 -1 0 -1 1
1 0 0 -1 0 0 -1 0 1 0 0 0 0 1 1 0 0 0 -1 0 0 -1 1 0 0 -1 -1 0 0 0 1 0
1 2 -1 -3 0 0 -1 -1 0 1 -1 0 -1 1 1 -1 0 -1 -1 1 0 -1 2 1 0 -2 0 2 0 -1 2 1
2 0 0 -2 1 0 0 -1 -1 0 -1 1 -1 0 1 0 0 -1 -1 0 1 -1 1 0 0 -1 0 1 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 -1 0 0 0 0 0 0 0 0 0
0 2 -1 -2 -1 -1 -1 0 1 1 0 0 0 1 1 -1 0 0 0 1 -1 -1 1 1 0 -2 1 1 0 0 1 0
2 1 0 -4 0 0 -1 -2 -1 1 -2 2 -2 1 2 -1 0 -2 -2 2 0 -2 2 1 1 -3 1 3 0 -2 3 2
1 2 -1 -3 -1 -1 -2 0 2 0 0 1 0 1 2 -1 0 0 -1 1 -1 -2 2 1 0 -3 0 1 0 0 2 0
1 0 1 1 -1 1 0 0 0 0 1 -2 0 0 -1 0 -1 1 1 0 0 -1 0 0 0 0 0 -2 1 0 0 -1
0 -1 1 3 0 1 2 1 -1 0 1 -2 0 -1 -3 0 0 0 2 -1 1 1 -1 -1 0 3 0 -2 0 0 -2 -1
1 0 0 0 1 0 2 0 -2 0 -1 0 -1 -1 -1 0 1 -2 0 -1 1 1 0 0 0 1 0 1 0 -1 0 1
1 4 -2 -5 -1 -1 -4 -1 2 1 -1 1 1 2 3 -2 1 0 -2 1 -1 -2 3 2 0 -5 0 3 -1 -1 3 1
1 -1 1 2 0 1 1 0 -1 0 1 -1 -1 -1 -2 1 -1 0 1 0 1 -1 0 -1 0 2 0 -2 1 0 -1 -1
-1 5 -4 -4 -1 -3 -4 1 4 0 0 1 2 2 3 -2 1 1 -1 1 -3 0 1 2 0 -5 0 3 -1 1 2 1
-
PROPERTIES
INTEGRAL = 1
MODULAR = 5
-
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
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe