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 13:14:11 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
  F4(C15:C4).2
  

• DIMENSION
  32
  

• GRAM (floating point or integer Gram matrix)
  32 0
  8
  1 8
  3 2 8
  2 3 2 8
  2 -1 1 -1 8
  2 0 2 1 2 8
  2 4 2 2 1 4 8
  4 1 4 2 2 2 2 8
  2 -1 1 1 4 1 2 2 8
  2 1 4 2 1 2 2 3 1 8
  1 1 2 2 2 0 0 1 2 2 8
  1 3 4 4 1 2 2 4 1 2 2 8
  3 -1 2 1 3 -1 -1 3 3 2 2 1 8
  2 2 1 0 1 2 2 2 1 1 4 0 2 8
  2 4 2 4 1 4 4 2 1 2 0 2 -1 2 8
  2 4 4 2 1 4 4 4 1 1 0 4 -1 2 4 8
  1 2 2 1 2 2 2 2 2 2 3 2 1 4 2 4 8
  1 1 2 1 2 1 2 1 4 2 1 1 4 2 1 1 2 8
  4 1 3 1 2 2 2 4 2 3 4 1 2 4 2 2 1 0 8
  0 1 4 1 2 1 1 2 2 2 1 2 4 2 1 2 4 4 0 8
  2 2 4 4 3 2 2 2 1 4 2 2 2 1 4 2 1 2 3 2 8
  1 2 1 2 1 1 1 1 2 1 4 2 2 3 2 1 2 2 2 2 0 8
  4 1 2 4 2 3 2 4 2 2 1 2 3 2 4 2 1 1 4 1 4 1 8
  1 2 1 1 2 2 4 1 4 1 3 1 1 4 2 2 4 4 1 2 1 2 1 8
  2 -1 2 1 4 1 1 4 4 0 2 2 3 1 1 2 4 2 2 4 1 2 2 2 8
  2 1 4 2 1 3 2 2 1 4 2 2 2 1 3 2 1 2 3 2 4 3 1 1 1 8
  2 3 2 2 0 -1 1 1 2 2 4 2 1 2 1 0 3 2 2 1 1 4 1 2 2 2 8
  1 2 1 2 2 2 2 1 2 1 3 1 1 4 4 2 4 2 1 2 2 1 2 4 2 0 2 8
  4 0 2 3 0 3 2 4 2 2 1 2 3 2 2 2 1 1 4 1 1 3 4 1 2 2 1 1 8
  2 2 4 2 1 1 2 3 1 4 2 2 2 2 2 1 2 2 2 2 4 1 2 1 0 4 3 1 1 8
  2 2 1 1 0 2 2 2 2 2 2 0 1 4 2 2 4 2 2 2 1 1 1 4 1 1 2 4 2 3 8
  2 0 1 3 0 0 1 2 4 1 2 1 3 1 1 1 2 2 2 2 -1 3 2 2 4 1 2 2 4 -1 1 8
  

• DIVISORS (elementary divisors)
  1^16 3^8 15^8
  

• MINIMAL_NORM
  8
  

• KISSING_NUMBER
  3600
  

• GROUP_ORDER
  2^10 * 3^3 * 5
  

• 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
  1 -2 -1 -1 -2 3 1 4 -1 -4 -1 1 2 -4 1 -3 4 0 4 1 5 2 -4 1 -5 -3 1 0 -3 0 -1 3
  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 0 0 0 0 0 -1 -1 1 0 0 0 -1 1 0 1 0 1 0 -1 0 -1 0 0 0 0 0 0 1
  0 0 0 -1 -1 1 0 2 0 -1 0 0 0 -1 0 -1 1 0 0 0 2 1 -1 0 -1 -1 0 0 0 0 0 1
  0 0 -1 0 0 0 0 1 0 0 0 0 -1 0 -1 0 0 0 0 1 1 0 0 0 -1 0 0 0 0 0 0 1
  2 -4 -4 -2 -4 4 1 4 -2 -4 -1 1 4 -6 2 0 3 -1 6 3 7 3 -6 3 -6 -6 2 0 -4 2 -2 4
  0 0 0 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
  1 0 -2 -1 -3 3 -1 1 0 -1 -1 0 2 -3 0 0 1 -1 3 1 3 2 -3 2 -1 -3 0 1 -1 2 -2 1
  0 0 0 0 0 0 0 1 0 -1 0 0 -1 0 0 -1 1 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 1
  0 0 0 0 -1 2 0 2 0 -2 -1 0 1 -2 0 -2 2 0 2 0 2 1 -2 0 -2 -1 0 1 -1 0 0 1
  0 0 0 1 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 -1 -1 1 0 1 0 -1 1 0 1 -1 2 -1 1 0 0 0 1 0 -1 0 -1 -1 0 -1 0 0 0 1
  0 0 0 0 -1 2 0 1 0 -1 0 0 1 -1 0 -1 1 0 1 0 1 0 -1 0 -1 -1 0 0 -1 0 0 1
  0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 1 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 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 0 1 0 0 0
  1 -2 -3 -1 -3 4 -1 2 -1 -2 0 0 4 -4 3 1 1 -1 4 2 4 1 -5 3 -3 -5 2 -1 -2 2 -2 2
  0 0 0 0 -1 1 0 1 0 -1 -1 0 0 -1 -1 -1 1 0 1 0 1 1 0 0 -1 0 0 1 -1 0 0 1
  0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 -1 0 0 -1 0 0 0 0 0 -1 1 0 0 0
  0 0 0 -1 -1 1 0 2 0 -2 -1 1 0 -1 0 -2 2 0 1 0 3 1 -1 0 -2 -1 0 0 -1 0 0 2
  0 0 0 0 0 1 0 1 0 -1 0 0 0 0 0 -1 0 0 0 0 1 0 -1 0 -1 0 0 0 0 0 0 1
  0 0 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
  1 -2 -3 -1 -3 4 0 2 -1 -2 0 0 4 -4 2 1 1 -1 4 2 4 1 -5 2 -3 -5 2 0 -2 2 -2 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 1 0 0 0
  0 0 0 0 0 0 0 1 0 -1 0 0 -1 0 0 -1 0 0 0 1 1 0 0 0 -1 0 0 0 0 0 0 1
  0 0 0 0 -1 2 0 2 0 -2 -1 0 1 -2 0 -2 2 0 2 0 2 1 -2 1 -2 -1 0 1 -1 0 -1 1
  -1 0 1 1 0 1 0 0 0 0 -1 0 1 0 0 -1 1 0 1 -1 0 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 0 0 0 -1 0 0 0 0 0 0 1
  1 -2 -2 -1 -2 2 0 2 -1 -2 0 1 2 -3 2 0 1 -1 3 2 3 1 -3 2 -3 -3 1 0 -2 1 -1 2
  0 -1 -1 0 -1 2 0 1 0 -1 0 0 2 -2 1 0 1 -1 2 1 1 0 -2 1 -2 -2 1 0 -1 1 -1 1
  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 0 0 0
  32
  -2 2 2 2 4 -4 -1 -6 0 5 2 -1 -2 6 -1 4 -6 0 -6 -2 -7 -4 5 -1 7 4 0 -1 4 0 2 -3
  0 2 0 0 0 0 -1 -2 1 2 0 0 0 2 -1 1 -2 -1 -2 -1 -1 -1 1 0 3 1 0 0 1 0 0 -1
  -2 4 3 3 4 -5 -2 -8 2 7 1 -1 -4 8 -3 4 -7 0 -7 -3 -9 -4 7 -2 10 6 -2 0 5 0 2 -5
  0 2 1 1 0 0 -1 -1 1 1 -1 0 -1 1 -2 -1 0 0 0 -1 -1 0 1 0 2 1 -1 1 0 0 0 -1
  -1 0 1 1 2 -2 0 -3 0 2 1 0 -1 3 0 2 -3 0 -3 -1 -3 -2 2 0 3 2 0 -1 2 0 1 -1
  -1 0 0 2 1 0 -1 -3 0 2 0 -1 1 1 0 3 -3 -1 0 0 -3 -1 0 1 3 0 1 0 1 1 0 -2
  0 0 -1 0 0 0 -1 -3 0 2 1 0 1 1 1 3 -3 -1 -1 0 -1 -1 0 1 3 -1 1 -1 1 1 0 -1
  -2 2 2 3 4 -4 -2 -7 0 6 1 -1 -2 6 -1 5 -7 0 -5 -2 -8 -4 5 0 8 4 0 -1 4 0 2 -4
  -1 0 1 1 2 -3 0 -4 0 3 1 0 -1 3 0 3 -3 0 -3 -1 -4 -2 3 0 4 2 0 -1 2 0 1 -2
  -2 4 4 2 4 -4 -1 -4 2 4 1 -1 -5 6 -3 0 -3 1 -6 -3 -6 -3 6 -3 6 6 -2 0 4 -2 2 -3
  -2 4 4 3 4 -4 0 -4 2 4 1 -1 -4 6 -3 0 -3 1 -6 -3 -7 -3 6 -3 6 6 -2 0 4 -2 2 -4
  -1 4 2 2 1 -1 -2 -3 2 3 -1 -1 -2 3 -3 0 -2 0 -2 -2 -4 -1 3 0 5 3 -2 1 2 0 0 -3
  -1 2 3 1 3 -4 0 -2 1 2 1 0 -4 4 -2 -1 -1 1 -5 -2 -4 -2 5 -2 3 5 -2 0 3 -2 2 -2
  -2 2 3 3 4 -3 0 -3 1 3 1 -1 -2 5 -1 1 -3 0 -5 -2 -6 -3 4 -2 4 4 0 -1 3 -2 2 -3
  0 0 -1 1 0 0 -1 -3 0 2 0 0 1 1 0 3 -3 -1 0 0 -2 -1 0 1 3 0 1 0 0 1 0 -1
  -1 0 -1 2 1 -1 -2 -6 0 5 1 -1 2 3 1 7 -7 -2 -2 0 -5 -3 1 2 6 0 2 -1 2 2 0 -3
  -2 2 3 2 3 -3 0 -3 2 3 1 -1 -2 5 -1 1 -3 0 -5 -2 -5 -3 4 -2 4 4 0 -1 3 -2 2 -3
  -1 1 2 1 2 -3 0 -2 1 2 0 0 -2 3 -1 0 -1 0 -3 -2 -3 -1 3 -1 3 3 -1 0 2 -1 1 -2
  -2 2 2 3 4 -4 -1 -7 0 6 2 -1 -2 6 -1 5 -7 0 -6 -2 -8 -4 5 -1 8 4 0 -1 4 0 2 -4
  -1 2 3 1 2 -3 0 -2 2 2 0 0 -3 4 -2 -1 -1 0 -4 -2 -3 -1 4 -2 3 4 -2 0 2 -2 2 -2
  -1 2 2 2 2 -2 -1 -4 1 3 0 0 -2 4 -2 1 -3 0 -3 -2 -4 -2 3 -1 5 3 -1 0 2 0 1 -2
  -1 2 2 2 2 -2 0 -3 1 3 1 -1 -1 3 -1 1 -2 0 -3 -2 -5 -2 3 -1 4 3 -1 0 2 -1 1 -3
  -1 1 1 2 2 -2 -1 -4 0 3 0 0 -1 3 -1 2 -3 0 -2 -1 -4 -2 2 0 4 2 0 0 2 0 1 -2
  -1 1 1 1 2 -2 0 -2 1 2 1 0 -1 3 0 1 -2 0 -3 -1 -3 -2 2 -1 3 2 0 -1 2 -1 1 -2
  -1 0 1 1 2 -3 0 -4 0 3 1 0 -1 3 0 3 -4 0 -3 -1 -4 -2 3 0 4 2 0 -1 2 0 2 -2
  -1 2 2 2 2 -2 -1 -4 1 3 1 -1 -1 3 -1 2 -3 0 -3 -2 -5 -2 3 -1 5 3 -1 0 2 0 1 -3
  -2 4 4 2 3 -3 0 -3 2 3 0 -1 -3 5 -3 -1 -1 1 -5 -4 -5 -2 5 -3 5 5 -2 0 3 -2 2 -3
  -1 1 1 1 1 -1 0 -1 1 1 0 0 -1 2 -1 0 -1 0 -2 -1 -2 -1 2 -1 2 2 0 0 1 -1 1 -1
  -1 1 1 2 2 -2 -1 -4 0 4 1 -1 -1 3 -1 3 -4 0 -3 -1 -5 -2 3 0 5 2 0 0 2 0 1 -3
  -2 4 5 3 4 -4 -1 -5 2 4 0 -1 -4 7 -3 0 -3 1 -6 -4 -7 -3 6 -3 7 6 -2 0 4 -2 2 -4
  -2 2 3 2 3 -3 0 -3 1 3 1 -1 -2 5 -1 1 -3 0 -5 -2 -5 -3 4 -2 5 4 0 -1 3 -2 2 -3
  0 0 0 0 1 -2 0 -2 0 2 1 0 -1 1 0 2 -2 0 -2 0 -2 -1 2 0 2 1 0 0 1 0 1 -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 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
  -2 4 4 3 4 -4 0 -4 2 4 1 -1 -4 6 -3 0 -3 1 -6 -3 -7 -3 6 -3 6 6 -2 0 4 -2 2 -4
  2 -2 -3 -2 -3 3 0 3 -1 -3 -1 1 2 -5 1 -1 3 0 5 2 5 3 -4 2 -4 -4 0 1 -3 2 -2 3
  1 -2 -1 -1 -1 0 1 1 0 -1 0 1 0 -1 1 0 1 0 1 1 2 1 -1 0 -2 -1 0 0 -1 0 0 1
  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 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 1 0 -1 -1 0 0 -1 -1 -1 1 0 1 0 1 1 0 0 -1 0 0 1 -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 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 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 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 -1 0 0 0 -1 0 0 0 0 0 1 1 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 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 2 1 0 0 -1 0 -1 1 1 0 0 -2 1 -2 -1 0 1 -1 -1 -1 0 2 -1 2 2 -2 1 1 0 0 -1
  0 -2 -1 0 0 1 1 1 -1 -1 1 0 2 -1 2 1 0 0 1 1 1 0 -2 0 -2 -2 2 -1 -1 0 0 1
  0 2 2 0 0 0 1 3 2 -2 -1 0 -2 0 -2 -5 4 1 0 -1 1 1 1 -2 -2 2 -2 1 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 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 1 0 0 0 0 0 0 0
  0 -2 -2 0 0 -1 0 -4 -1 3 2 0 2 1 2 6 -5 -1 -1 1 -2 -2 0 1 3 -1 2 -1 1 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 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 1 0 0 0 0 0 0 0 0
  -1 0 2 1 3 -3 1 -1 0 1 1 0 -2 3 0 0 -1 1 -3 -1 -3 -2 3 -2 1 3 0 -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 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
  -2 2 3 2 4 -3 0 -3 1 3 1 -1 -2 5 -1 1 -3 0 -5 -2 -5 -3 4 -2 4 4 0 -1 3 -2 2 -2
  

• PROPERTIES
  INTEGRAL = 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
  
  

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  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe