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 13:14:36 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
  SL2(19)SL2(3).2
  

• DIMENSION
  36
  

• GRAM (floating point or integer Gram matrix)
  36 0
  10
  3 10
  4 3 10
  4 2 0 10
  2 0 0 3 10
  0 2 4 2 4 10
  2 4 3 0 2 3 10
  3 4 2 3 4 0 0 10
  0 0 2 4 3 3 4 2 10
  3 3 0 0 2 -1 3 2 0 10
  3 4 4 1 -1 2 0 0 -1 3 10
  4 3 2 3 -1 1 1 0 -1 3 4 10
  4 1 0 4 4 0 0 1 0 0 1 1 10
  4 0 4 1 3 3 1 0 0 -1 2 2 4 10
  3 1 3 0 1 2 1 0 -3 1 4 4 3 3 10
  3 3 2 3 -1 2 0 0 0 1 4 4 3 2 4 10
  2 3 0 0 3 1 3 1 1 4 3 -1 2 3 0 -1 10
  2 1 1 0 3 4 3 0 0 4 3 3 -1 3 4 1 4 10
  4 3 3 0 0 2 4 -1 -1 3 4 1 1 4 2 4 4 3 10
  5 3 2 4 1 0 1 2 0 2 3 4 4 3 1 4 1 1 4 10
  3 0 3 1 3 3 2 -1 0 0 4 1 2 5 4 1 4 3 4 3 10
  5 2 2 2 1 0 1 1 0 3 4 4 4 1 2 3 3 1 2 5 4 10
  4 1 4 0 0 0 0 0 1 2 4 4 1 4 3 2 3 3 1 2 4 4 10
  0 1 3 0 2 3 2 -1 0 1 2 1 3 3 4 1 0 1 2 -1 2 -1 0 10
  0 1 3 0 4 3 0 4 1 1 1 0 1 3 1 0 2 3 1 1 2 1 1 2 10
  3 2 1 4 3 1 -1 4 -1 0 0 3 4 1 2 1 -1 0 -2 2 1 3 0 2 4 10
  -1 1 3 0 3 3 0 3 2 1 1 -1 1 2 1 -1 3 3 0 0 1 -1 1 1 5 1 10
  2 5 1 1 0 1 4 0 0 3 3 3 4 0 1 1 2 -1 2 1 0 3 0 4 0 2 -1 10
  1 4 1 2 0 0 0 5 1 3 3 4 1 -1 1 0 1 1 -2 1 -1 2 1 1 3 3 2 4 10
  2 4 3 0 1 1 1 3 1 4 2 1 1 0 0 1 3 2 0 0 0 4 3 1 4 4 4 2 2 10
  1 3 0 1 2 1 0 3 -3 4 3 3 1 0 4 1 1 3 1 0 0 1 0 3 2 1 1 2 4 1 10
  1 2 0 4 1 -1 1 1 1 2 1 0 4 0 0 1 2 -4 1 1 1 2 -1 3 -1 3 0 4 1 1 1 10
  1 0 1 1 1 1 1 3 1 3 2 2 -1 0 3 0 0 4 0 -1 0 0 1 1 3 1 4 0 4 2 3 0 10
  4 3 1 4 1 0 0 2 1 1 0 0 5 2 0 4 1 -1 2 2 -2 0 -1 3 0 2 1 3 1 2 1 3 -1 10
  3 2 4 1 0 0 3 1 0 3 2 4 1 1 2 2 -1 0 2 1 -1 0 0 4 0 2 -1 4 2 0 3 3 0 4 10
  2 -1 2 3 2 3 2 0 1 -2 1 0 3 2 4 1 1 2 1 1 4 1 1 1 0 2 2 1 -1 -1 1 0 1 1 1 10
  

• DIVISORS (elementary divisors)
  1^18 2^10 38^8
  

• MINIMAL_NORM
  10
  

• KISSING_NUMBER
  4104
  

• GROUP_ORDER
  2^6 * 3^3 * 5 * 19
  

• GROUP_GENERATORS
  2
  36
  0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
  6 6 -15 0 -4 6 5 6 -7 2 5 2 3 2 -3 -5 -12 1 -1 -6 -3 7 3 6 9 -13 2 -11 3 1 -10 7 -5 -1 4 8
  9 10 -27 4 -5 9 8 6 -11 2 7 -3 1 6 -1 -7 -17 -2 -1 -7 -6 13 6 10 12 -19 6 -16 6 3 -15 7 -7 -4 10 10
  7 6 -18 2 -3 6 6 5 -8 1 5 -1 1 3 -2 -4 -11 -1 -1 -5 -4 8 4 8 8 -13 4 -11 4 2 -10 5 -5 -3 6 7
  11 11 -30 2 -7 11 10 10 -13 3 9 1 4 5 -4 -9 -21 0 -2 -10 -6 14 6 12 16 -24 5 -20 6 3 -18 11 -9 -3 9 13
  11 12 -30 4 -6 11 8 7 -12 3 7 -2 2 6 -2 -8 -19 -2 -1 -9 -7 15 6 12 14 -22 6 -19 7 3 -17 8 -8 -5 11 12
  2 0 -5 2 1 1 3 0 -3 -1 2 -2 -2 1 0 0 -2 -2 0 0 -2 2 2 2 1 -2 2 -2 2 1 -2 0 -1 -1 2 1
  10 9 -29 5 -5 10 10 7 -13 2 8 -3 0 6 -1 -7 -18 -3 -1 -8 -7 14 7 11 13 -20 7 -17 7 3 -16 7 -8 -4 10 10
  -12 -13 34 -4 7 -12 -10 -9 14 -3 -9 2 -2 -7 2 10 22 2 1 10 8 -17 -7 -13 -16 25 -7 21 -7 -3 20 -10 9 5 -12 -13
  5 7 -15 1 -3 5 5 5 -7 2 5 1 2 3 -2 -5 -12 1 -1 -6 -4 8 3 6 9 -13 2 -11 3 1 -10 7 -5 -1 4 8
  2 4 -6 -2 -2 3 1 3 -2 1 2 3 4 0 -2 -3 -6 2 0 -3 -1 2 1 2 5 -7 -1 -6 1 1 -5 5 -2 0 1 5
  4 3 -12 4 -3 4 5 3 -6 1 3 -3 0 3 0 -2 -6 -2 -1 -3 -3 6 3 5 4 -7 4 -6 2 1 -6 1 -3 -3 5 3
  3 2 -10 2 -3 4 5 4 -5 1 3 -1 0 2 -1 -2 -6 -1 -1 -3 -2 5 3 4 4 -6 3 -5 1 0 -5 2 -3 -1 3 3
  11 14 -33 4 -7 12 10 9 -14 4 9 -2 4 7 -3 -10 -23 -1 -1 -11 -8 17 7 13 16 -25 6 -22 7 3 -20 10 -9 -5 12 14
  -10 -11 24 -2 3 -8 -5 -4 9 -2 -5 2 0 -5 1 6 15 2 1 6 6 -12 -4 -10 -11 18 -5 15 -6 -3 14 -7 6 5 -9 -10
  6 6 -17 5 -2 5 5 2 -7 1 3 -5 -3 5 1 -3 -9 -3 -1 -3 -5 9 4 7 6 -10 5 -8 4 2 -8 2 -4 -4 7 5
  1 2 -5 0 -1 2 2 2 -2 1 2 1 1 1 -1 -2 -4 0 0 -2 -1 2 1 2 3 -4 0 -4 1 1 -3 2 -2 0 1 3
  -6 -6 15 -1 2 -5 -4 -3 6 -1 -4 1 -1 -3 1 4 10 1 0 4 4 -7 -3 -6 -7 11 -3 10 -4 -2 9 -5 4 3 -5 -6
  -2 -3 5 0 1 -2 -1 -1 2 -1 -1 0 -1 -1 1 2 4 0 0 2 1 -3 -1 -2 -3 4 -1 4 -1 0 3 -2 1 1 -2 -2
  22 26 -63 7 -12 22 18 16 -26 6 16 -4 4 13 -4 -18 -42 -3 -2 -19 -15 32 13 25 30 -47 12 -40 14 6 -37 19 -17 -9 22 26
  7 9 -20 2 -4 7 5 5 -8 2 5 -1 2 4 -1 -6 -13 -1 -1 -6 -5 10 4 8 10 -15 3 -13 4 2 -12 6 -5 -3 7 9
  3 3 -10 2 -2 4 4 3 -5 1 3 -1 0 2 0 -3 -7 -1 0 -3 -3 5 3 4 5 -7 2 -6 2 1 -6 3 -3 -1 3 4
  8 10 -23 0 -6 9 7 9 -10 3 7 3 6 3 -4 -8 -18 2 -1 -9 -4 11 4 9 13 -20 3 -17 4 2 -15 10 -7 -2 7 11
  -6 -8 17 -2 3 -6 -3 -3 6 -2 -3 2 -1 -4 0 5 11 1 0 5 4 -9 -3 -6 -8 13 -3 11 -4 -2 10 -5 4 3 -7 -7
  13 15 -36 2 -7 13 10 10 -15 3 10 0 5 6 -4 -11 -25 0 -1 -11 -8 17 7 14 18 -28 6 -24 8 4 -22 13 -10 -5 12 16
  3 1 -7 3 -1 2 4 1 -4 0 2 -3 -2 2 0 0 -3 -2 -1 -1 -2 3 2 3 2 -3 3 -3 2 1 -3 0 -2 -2 3 1
  6 7 -14 1 -3 5 3 4 -6 1 3 1 3 2 -2 -4 -10 1 -1 -5 -3 7 2 6 8 -13 2 -10 3 2 -9 6 -4 -3 5 7
  1 1 -3 0 0 1 2 1 -2 0 2 1 1 0 -1 -1 -3 0 0 -1 -1 1 1 1 2 -3 0 -3 1 1 -2 2 -1 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 1 0 0 0 0 0 0 0 0 0 0
  -6 -6 16 -1 3 -6 -4 -5 7 -1 -4 0 -1 -3 1 5 11 0 0 5 4 -8 -3 -6 -8 13 -3 10 -4 -2 10 -6 5 2 -5 -7
  17 19 -47 7 -9 16 13 10 -19 4 11 -6 2 10 -2 -12 -30 -3 -2 -13 -11 24 10 18 21 -33 11 -28 10 4 -26 12 -12 -8 18 17
  8 9 -23 3 -4 8 8 5 -10 2 7 -2 1 5 -2 -6 -15 -2 -1 -7 -6 12 5 9 11 -16 5 -15 5 2 -13 6 -6 -3 8 9
  -11 -13 30 -2 5 -11 -8 -7 12 -3 -8 1 -2 -6 2 9 20 2 1 9 8 -15 -6 -12 -15 23 -5 20 -7 -3 18 -10 8 4 -10 -13
  4 5 -8 -2 -2 3 1 3 -3 1 2 3 4 0 -3 -3 -8 3 0 -4 -1 3 1 3 6 -9 0 -7 2 1 -6 6 -3 -1 2 6
  18 20 -50 7 -9 17 14 11 -21 4 12 -5 2 11 -3 -13 -32 -3 -2 -14 -12 25 10 20 23 -37 10 -31 12 6 -29 14 -13 -9 19 20
  36
  -15 -19 43 -5 7 -15 -11 -10 18 -4 -10 3 -3 -9 3 12 29 2 1 13 10 -22 -9 -17 -20 33 -8 28 -10 -5 26 -13 11 7 -16 -18
  -21 -25 59 -5 10 -21 -16 -15 25 -5 -15 1 -5 -11 5 18 40 2 1 18 14 -29 -12 -23 -29 46 -10 39 -14 -7 36 -20 16 8 -20 -26
  7 7 -17 1 -4 6 5 5 -7 2 5 0 2 3 -2 -5 -12 0 -1 -6 -4 9 3 7 9 -13 4 -11 3 0 -10 6 -5 -2 5 7
  -9 -12 26 -1 4 -9 -7 -7 11 -2 -7 -1 -3 -5 3 8 19 0 0 9 6 -13 -5 -10 -13 21 -4 18 -6 -3 16 -10 7 3 -8 -12
  5 5 -17 4 -3 6 6 4 -8 2 5 -2 0 4 0 -5 -11 -2 0 -5 -5 9 4 6 8 -12 4 -10 4 2 -10 4 -5 -2 6 6
  13 13 -36 6 -6 12 11 8 -15 3 9 -5 0 8 -1 -9 -23 -3 -1 -10 -10 19 8 14 16 -25 9 -21 8 3 -20 9 -10 -6 13 13
  6 7 -16 1 -4 6 5 5 -7 2 5 0 3 3 -2 -5 -12 1 -1 -6 -4 8 3 6 9 -13 3 -11 3 1 -10 6 -5 -2 5 7
  -15 -18 41 -3 6 -14 -11 -10 17 -3 -10 1 -3 -8 3 12 28 1 1 12 10 -20 -8 -16 -20 32 -7 27 -10 -5 25 -14 11 6 -14 -18
  7 9 -21 2 -5 8 6 6 -9 3 6 -1 3 4 -2 -7 -15 0 -1 -7 -5 11 5 8 11 -16 4 -14 4 1 -13 7 -6 -2 7 9
  -9 -10 22 -2 3 -7 -5 -5 9 -1 -5 1 -1 -4 2 6 15 0 1 6 5 -11 -4 -9 -11 18 -4 14 -5 -3 13 -8 6 4 -8 -10
  -3 -3 8 -1 1 -3 -2 -2 4 0 -2 0 -1 -1 1 2 5 0 0 2 2 -3 -2 -3 -4 7 -1 5 -2 -2 5 -3 2 2 -3 -4
  -2 -4 8 -1 1 -3 -2 -2 4 -1 -2 0 -1 -2 1 3 6 0 0 3 2 -5 -2 -3 -4 7 -1 6 -2 -1 5 -3 2 1 -3 -4
  -3 -3 7 0 1 -3 -2 -2 3 0 -2 0 -1 -1 1 2 5 0 0 2 2 -3 -2 -3 -3 6 -1 5 -2 -1 4 -3 2 1 -2 -3
  1 0 -1 -1 0 0 1 1 -1 0 1 2 1 -1 -1 -1 -2 1 0 -1 0 0 0 0 2 -2 0 -1 0 0 -1 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 -1 0 0 0 0 0 0 0 0 0 0 0 0 0
  -15 -17 42 -5 7 -15 -12 -10 18 -3 -11 2 -3 -8 3 12 28 2 1 13 10 -21 -9 -16 -20 32 -8 27 -10 -5 25 -13 11 6 -15 -17
  -13 -14 35 -4 6 -12 -9 -8 14 -2 -8 3 -1 -7 2 9 22 2 1 9 8 -17 -7 -14 -16 26 -7 21 -8 -4 20 -10 9 6 -13 -14
  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
  -2 -3 7 -3 0 -2 -1 0 3 0 -1 3 2 -2 -1 1 3 2 0 1 2 -4 -2 -3 -2 4 -2 3 -2 -1 4 0 1 2 -4 -2
  1 -1 1 -1 0 -1 0 -1 1 0 0 0 -1 0 0 1 1 0 -1 1 0 -1 -1 0 0 1 0 1 0 0 1 0 0 0 -1 0
  8 8 -21 3 -3 7 7 4 -9 2 6 -2 0 5 -1 -6 -14 -2 -1 -6 -6 11 4 8 10 -15 5 -13 5 2 -12 6 -6 -3 7 8
  -7 -9 19 -1 3 -7 -4 -5 8 -1 -4 0 -3 -3 2 6 14 -1 0 6 4 -9 -4 -7 -10 17 -2 13 -5 -3 12 -8 5 3 -7 -9
  -9 -9 26 -5 5 -9 -8 -6 11 -2 -6 4 0 -6 1 6 16 2 1 7 6 -13 -6 -10 -11 18 -6 15 -6 -3 14 -6 7 5 -10 -9
  2 3 -7 0 -2 3 2 3 -3 1 2 1 2 1 -1 -3 -6 1 0 -3 -1 4 1 2 4 -6 1 -5 1 0 -5 3 -2 0 2 3
  1 1 -2 -1 -1 1 0 2 -1 1 1 2 2 0 -2 -1 -3 1 0 -2 0 1 0 1 2 -3 0 -2 0 0 -2 2 -1 0 0 2
  -12 -15 34 -3 6 -12 -9 -8 14 -3 -8 2 -3 -7 2 10 23 1 1 10 8 -17 -7 -13 -16 26 -6 22 -8 -4 20 -11 9 5 -12 -14
  13 15 -37 5 -7 13 10 8 -15 4 9 -4 2 8 -2 -10 -24 -3 -1 -11 -9 19 8 14 17 -26 8 -23 8 3 -21 10 -9 -6 14 14
  2 3 -7 1 -2 2 2 2 -2 1 2 -1 1 2 0 -2 -5 0 0 -3 -1 4 1 2 3 -4 2 -4 1 0 -4 1 -2 -1 3 2
  -4 -5 11 0 1 -4 -3 -3 5 -1 -3 -1 -1 -2 1 4 8 0 0 3 3 -5 -2 -4 -6 10 -1 8 -3 -2 7 -5 3 1 -3 -6
  -15 -16 38 -3 6 -13 -10 -9 16 -2 -9 1 -3 -7 3 11 26 0 1 11 9 -18 -7 -15 -19 31 -6 25 -10 -6 23 -14 11 6 -13 -17
  -10 -12 26 1 4 -9 -7 -8 11 -2 -8 -3 -4 -4 4 9 20 -2 1 9 6 -12 -5 -10 -15 23 -3 19 -6 -3 17 -13 8 2 -7 -14
  -15 -18 41 -3 7 -14 -10 -10 16 -3 -10 1 -3 -8 3 12 28 1 1 12 10 -20 -8 -16 -20 32 -7 26 -9 -5 24 -14 11 6 -14 -18
  4 5 -11 -1 -3 5 2 4 -4 1 3 1 3 1 -2 -4 -8 1 0 -4 -2 5 2 4 6 -9 1 -8 2 1 -7 5 -3 -1 4 5
  -15 -17 40 -4 6 -14 -11 -9 17 -3 -10 2 -2 -8 3 11 27 2 1 12 10 -20 -8 -16 -19 31 -8 26 -10 -5 24 -13 11 6 -14 -17
  1 1 -2 0 -2 1 1 2 -1 1 1 1 2 0 -1 -1 -2 1 0 -2 0 1 0 1 2 -2 0 -2 0 -1 -2 1 -1 0 0 1
  18 22 -49 5 -9 17 12 11 -19 5 12 -3 4 10 -3 -15 -33 -2 -1 -15 -12 25 9 19 24 -37 9 -32 11 5 -29 15 -13 -8 18 21
  
  

• PROPERTIES
  INTEGRAL = 1
  

• REFERENCES
  G. Nebe, Finite quaternionic matrix groups,
  Representation Theory 2, 106-223 (April 1998)
  

• NOTES
  subgroup SL2(19) acts on
  2-modular lattice
  and
  2-modular lattice
  

• URL (links to other sites for this lattice)
  All the quaternionic matrix groups
  

• LAST_LINE
  

  ---

  Haftungsausschluss/Disclaimer
  
  
  Gabriele Nebe