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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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10 Hilbert basis elements of degree 1
10 extreme rays
22 support hyperplanes

embedding dimension = 6
rank = 6 (maximal)
external index = 3
internal index = 1

size of partial triangulation   = 1
resulting sum of |det|s = 2

grading:
1 1 1 1 1 1 
with denominator = 3

degrees of extreme rays:
1: 10  

***********************************************************************

10 Hilbert basis elements of degree 1:
 0 0 1 1 0 1
 0 0 1 1 1 0
 0 1 0 0 1 1
 0 1 0 1 1 0
 0 1 1 0 0 1
 1 0 0 0 1 1
 1 0 0 1 0 1
 1 0 1 0 1 0
 1 1 0 1 0 0
 1 1 1 0 0 0

10 extreme rays:
 0 0 1 1 0 1
 0 0 1 1 1 0
 0 1 0 0 1 1
 0 1 0 1 1 0
 0 1 1 0 0 1
 1 0 0 0 1 1
 1 0 0 1 0 1
 1 0 1 0 1 0
 1 1 0 1 0 0
 1 1 1 0 0 0

22 support hyperplanes:
 -2  1  1  1  1  1
 -1 -1  2  2 -1  2
 -1 -1  2  2  2 -1
 -1  2 -1 -1  2  2
 -1  2 -1  2  2 -1
 -1  2  2 -1 -1  2
  0  0  0  0  0  1
  0  0  0  0  1  0
  0  0  0  1  0  0
  0  0  1  0  0  0
  0  1  0  0  0  0
  1 -2  1  1  1  1
  1  0  0  0  0  0
  1  1 -2  1  1  1
  1  1  1 -2  1  1
  1  1  1  1 -2  1
  1  1  1  1  1 -2
  2 -1 -1 -1  2  2
  2 -1 -1  2 -1  2
  2 -1  2 -1  2 -1
  2  2 -1  2 -1 -1
  2  2  2 -1 -1 -1

1 congruences:
 1 1 1 1 1 1 3

6 basis elements of lattice:
 1 0 0 0 0 -1
 0 1 0 0 0 -1
 0 0 1 0 0 -1
 0 0 0 1 0 -1
 0 0 0 0 1 -1
 0 0 0 0 0  3