The Lattice Sp4(3)Sp4(3).2xC3.2
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 13:13:46 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
Sp4(3)Sp4(3).2xC3.2
-
DIMENSION
32
-
GRAM (floating point or integer Gram matrix)
32 0
6
0 6
-1 0 6
0 -1 1 6
0 0 0 -1 6
2 0 0 2 -3 6
0 0 -2 0 0 -1 6
-2 0 0 2 0 0 1 6
3 0 1 0 0 1 0 -1 6
0 -3 -1 -1 0 0 0 0 0 6
-1 1 -3 -2 0 -1 1 0 -3 0 6
0 1 2 3 1 1 -1 1 0 -3 -3 6
0 0 0 -1 -3 1 0 -1 0 0 0 -3 6
-1 0 1 -1 2 -3 -1 0 0 0 0 0 -3 6
0 0 -1 1 0 1 3 2 0 0 0 0 0 -3 6
1 0 0 -1 1 0 -2 -3 0 0 0 0 0 0 -3 6
-3 0 2 0 0 -1 0 1 0 0 0 0 0 0 0 0 6
0 -3 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 6
2 -1 -3 1 0 1 1 0 0 0 0 0 0 0 0 0 -3 0 6
0 2 1 -3 2 -1 -1 -1 0 0 0 0 0 0 0 0 0 -3 -3 6
0 0 0 2 -3 2 0 1 0 0 0 0 0 0 0 0 0 0 0 -3 6
-1 0 -1 -1 1 -3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 6
0 0 1 1 0 2 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 6
1 0 0 -1 -1 0 1 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 6
-3 0 -1 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 2 0 -2 6
0 3 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 1 0 0 0 0 0 6
1 -1 3 2 0 1 -1 0 0 0 0 0 0 0 0 0 1 2 0 -1 0 -2 2 0 -3 0 6
0 -1 -2 -3 -1 -1 1 -1 0 0 0 0 0 0 0 0 0 -1 1 0 -1 2 -2 2 0 -3 -3 6
0 0 0 1 3 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 -2 0 0 0 -3 6
1 0 -1 1 -2 3 1 0 0 0 0 0 0 0 0 0 -2 0 2 -2 1 0 1 0 0 0 0 0 -3 6
0 0 1 -1 0 -1 -3 -2 0 0 0 0 0 0 0 0 0 0 -2 2 0 -1 0 1 0 0 0 0 0 -3 6
-1 0 0 1 -1 0 2 3 0 0 0 0 0 0 0 0 2 0 0 -2 2 0 -1 0 0 0 0 0 0 0 -3 6
-
DIVISORS (elementary divisors)
1^16 3^16
-
MINIMAL_NORM
6
-
KISSING_NUMBER
9600
-
GROUP_ORDER
2^15 * 3^9 * 5^2
-
GROUP_GENERATORS
3
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 1 1 0 0 -1 -1 -1 -2 -1 -1 -1 -1 -1 -1 -1 -2 -1 -1 -1 -1 0 0 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 1 1 0 0 -1 -1 -1 -2 -1 -1 -1 -1
0 1 1 2 1 1 1 1 0 0 0 -2 0 0 -1 0 0 0 0 0 0 0 0 1 0 -1 -1 0 -1 -1 0 -1
2 2 2 4 4 4 2 2 -1 1 1 -2 2 2 -1 1 1 1 1 2 2 2 1 1 1 -1 -1 2 -2 -2 1 -1
-1 -1 -1 -2 -2 -2 -1 -1 0 -1 -1 0 -2 -2 0 -1 0 0 0 0 0 0 0 0 0 1 1 0 2 2 0 1
0 0 0 0 0 0 0 1 0 -1 -1 -2 -1 -1 -1 0 0 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0
1 1 1 2 1 1 1 1 -1 0 0 -2 0 0 -1 0 1 1 1 2 1 1 1 1 0 -1 -1 0 -1 -1 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 0
0 0 0 0 0 0 1 0 1 2 2 4 3 2 1 1 1 2 2 4 3 2 2 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 0 1 2 2 4 3 2 2 1
1 2 2 4 3 2 1 1 -1 0 0 -4 0 0 -1 0 0 0 0 0 0 0 -1 0 0 -2 -2 0 -3 -2 0 -1
-2 -4 -4 -8 -6 -4 -2 -2 1 -2 -2 4 -3 -2 1 -1 -1 -2 -2 -4 -3 -2 -1 -1 -1 2 2 -4 3 2 -1 1
1 2 2 4 3 2 1 1 0 2 2 0 3 2 0 1 0 0 0 0 0 0 0 0 0 -2 -2 0 -3 -2 0 -1
0 0 0 0 0 0 -1 0 -1 -2 -2 -4 -3 -2 -1 -1 1 2 2 4 3 2 1 1 0 0 0 0 0 0 -1 0
-1 -2 -2 -4 -3 -2 -2 -1 1 0 0 4 0 0 2 0 -1 -2 -2 -4 -3 -2 -2 -1 0 2 2 0 3 2 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 -1 -1 -2 -2 -1 -1 -1 -1
1 1 2 2 1 1 1 1 0 1 1 -1 1 1 0 1 1 1 1 1 1 1 1 1 0 -1 -2 0 -1 -1 0 -1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1
0 0 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -2 -2 -1 -1 -1 -1 0 0 1 0 0 0 0 0
0 0 2 2 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 1 0 0 0 0
0 0 -1 -1 0 0 0 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
-1 -1 -2 -2 -1 -1 -1 -1 0 -1 -1 1 -1 -1 0 -1 0 0 -1 -1 0 0 0 0 -1 0 0 -2 0 0 -1 0
-1 -1 -1 -1 -1 -1 -1 -1 1 0 0 1 0 0 1 0 -1 -1 -1 -1 -1 -1 -1 -1 0 1 1 0 1 1 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 -1 -1 -1 -1
0 0 0 0 1 1 1 1 0 0 0 0 1 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 -1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0
0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 1 0 1
0 0 0 0 0 2 2 2 0 0 0 0 0 1 -1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 -1 1 -1
0 0 0 0 0 -1 -1 -1 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 -1 -1 -1 -1 0 0 0 0 -1 0 1 0 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0
0 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 1 0 0 0
32
-4 -6 -8 -12 -10 -8 -6 -4 2 -3 -4 6 -5 -4 3 -2 -2 -3 -4 -6 -5 -4 -3 -2 -2 3 4 -6 5 4 -3 2
0 0 0 1 1 0 0 0 0 0 0 -1 0 0 0 0 -1 -1 -2 -2 -1 -1 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 1 0 0 0 0 0 0 -1 0 -1 -1 -1 -2 -2 -1 0 0 0 0 0 0 0 0 1 0
-2 -2 -2 -4 -2 0 0 0 1 -1 -1 2 -1 0 0 0 -1 -1 -1 -2 -1 0 0 0 -1 1 1 -2 1 0 0 0
1 2 2 3 2 1 1 0 -1 0 0 -3 0 0 -1 0 1 1 2 2 2 1 1 0 1 0 0 3 0 0 1 0
-2 -3 -3 -5 -4 -3 -2 -1 2 0 0 5 0 0 2 0 -2 -2 -3 -4 -3 -2 -2 -1 -2 0 0 -5 0 0 -2 0
2 2 2 4 4 4 2 0 -1 1 1 -2 2 2 -1 0 1 1 1 2 2 2 1 0 1 -1 -1 2 -2 -2 1 0
1 2 3 4 4 3 2 1 -1 0 0 -4 0 0 -2 0 1 1 2 3 3 3 2 1 1 0 0 4 0 0 2 0
0 0 0 0 0 0 0 0 -1 -1 -2 -3 -2 -2 -2 -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 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 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 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 2 2 3 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 -2 -3 -3 -5 -4 -3 -2 -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 0 0 0 0 0 0 0 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 3 4 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 4 6 9 7 6 5 3 -1 3 4 -3 5 4 -2 2 1 1 2 3 2 2 2 1 1 -3 -4 3 -5 -4 2 -2
-1 -1 -2 -4 -3 -1 0 0 0 -1 -2 1 -2 -1 0 0 0 0 0 -1 -1 0 0 0 0 1 2 -1 2 1 0 0
-1 -1 -1 -2 -2 -1 -2 -2 0 -1 -1 0 -2 -1 1 -1 0 0 0 0 0 0 -1 -1 0 1 1 0 2 1 -1 1
1 1 1 2 1 -1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 -1 -1 0 -1 0 0 0
-1 -3 -2 -4 -2 -1 -1 0 1 -1 0 3 0 0 1 0 -1 -2 -2 -3 -2 -1 -1 0 -1 1 0 -3 0 0 -1 0
2 4 3 6 5 4 2 1 -2 1 0 -5 1 1 -2 0 2 3 3 5 4 3 2 1 2 -1 0 5 -1 -1 2 0
-2 -1 -1 -2 -2 -2 -1 0 1 -1 -1 0 -2 -2 0 0 -1 0 0 0 0 0 0 0 -1 1 1 0 2 2 0 0
-1 -3 -4 -5 -5 -3 -2 -1 1 -1 -1 4 -1 0 2 0 -1 -2 -3 -4 -4 -3 -2 -1 -1 1 1 -4 1 0 -2 0
3 5 6 9 8 6 4 3 -2 2 2 -6 3 2 -3 1 3 5 6 9 8 6 4 3 1 -3 -4 3 -5 -4 1 -2
-2 -2 -4 -5 -3 -2 0 0 1 -1 -2 3 -1 -1 0 0 -2 -2 -4 -5 -3 -2 0 0 -1 1 2 -2 2 1 0 0
-2 -2 -2 -4 -4 -2 -1 -1 1 -1 -1 2 -2 -1 1 0 -2 -2 -2 -4 -4 -2 -1 -1 -1 1 1 -2 2 1 0 1
2 2 2 4 2 1 0 0 -1 1 1 -2 1 1 0 0 2 2 2 4 2 1 0 0 1 -1 -1 2 -1 0 0 0
1 0 2 1 2 1 1 0 0 1 2 1 2 1 0 0 1 0 2 1 2 1 1 0 1 1 0 2 0 0 1 0
-2 -1 -3 -3 -2 -1 -2 -1 0 -2 -3 -1 -3 -2 0 -1 -2 -1 -3 -3 -2 -1 -2 -1 -2 -1 0 -4 -1 -1 -2 0
-1 -2 -2 -4 -4 -4 -2 0 1 -1 -1 2 -2 -2 1 0 -1 -2 -2 -4 -4 -4 -2 0 0 1 1 -2 2 2 -1 0
1 0 1 2 2 3 2 1 0 1 2 1 3 3 0 1 1 0 1 2 2 3 2 1 1 1 1 3 1 0 2 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 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 -2 -4 -3 -2 -2 -1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -2 -1 -1 -1 -1 0 0 0 0 0 0 0 -1
1 1 2 3 3 2 1 1 0 1 2 0 3 2 0 1 0 0 0 0 0 0 0 0 0 -1 -2 0 -3 -2 0 -1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 4 5 4 3 2 1 2 3 4 5 4 3 2 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 -1 -1 -1 -1 -1 -1 -2 -2 -3 -4 -4 -3 -2 -1
0 0 0 0 0 -1 -1 -1 0 0 0 0 0 -1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 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 1 1 2 2 2 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 2 2 4 3 2 2 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 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
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 -3 -4 -5 -4 -3 -2 -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 2 2 3 4 4 3 2 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 -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 1 1 1 0 0 0
0 1 1 2 2 2 1 0 -1 0 0 -2 0 0 -1 0 1 1 1 2 2 2 1 0 0 0 0 2 0 0 1 0
1 2 2 4 3 2 2 1 0 2 2 0 3 2 -1 1 0 0 0 0 0 0 1 0 1 0 0 4 0 0 2 0
0 0 0 0 0 0 0 1 0 -1 -1 -2 -1 -1 -1 0 0 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 -1 -1 -1 -1 -1 -1 -1 -2 -2 -3 -2 0 -1 1 1 2 2 2 1 0 0 0 0 0 -1 0 0 -1 0
-2 -3 -4 -5 -4 -3 -2 -1 2 -1 0 5 0 0 2 0 -2 -2 -4 -5 -4 -3 -2 -1 -2 0 0 -5 0 0 -2 0
2 2 3 4 4 3 2 1 -1 2 2 -1 3 2 -1 0 1 0 1 1 1 1 1 1 2 0 0 4 0 0 2 0
0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0
0 0 1 1 1 0 0 0 1 1 2 1 1 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0
-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -2 -2 -2 -1 0 0 0 0 0 0 0 0 0
1 2 2 4 3 2 1 1 0 2 2 0 3 2 0 1 0 0 0 0 0 0 -1 0 0 -2 -2 0 -3 -2 0 -1
0 -1 -1 -2 -1 -1 -1 0 0 -1 -1 0 -1 -1 0 0 0 -1 -1 -2 -1 -1 -1 -1 0 1 1 0 1 1 0 0
-1 -1 -2 -3 -3 -2 -1 -1 0 -1 -2 0 -3 -2 0 -1 -1 -1 -2 -2 -2 -1 0 0 0 1 2 0 3 2 0 1
0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 2 2 4 5 4 3 2 1 0 1 0 0 0 0 0 0
1 2 2 3 3 2 1 0 0 2 2 0 3 2 0 0 -1 0 -1 -1 -1 -1 -1 -1 0 -2 -2 0 -3 -2 0 0
0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 -1 0 -1
1 1 2 2 1 0 0 0 0 1 2 0 1 0 0 0 1 1 1 1 0 0 0 0 0 -1 -2 0 -1 0 0 0
-
PROPERTIES
INTEGRAL = 1
MODULAR = 3
-
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