# p31m
#
http://en.wikipedia.org/wiki/Wallpaper_group#Group_p31m
M := [ [1,2,3], [1,2,4], [1,3,4], [2,3,6], [2,5,6], [2,4,7], [2,5,7], [3,4,7], [3,6,7] ];
S3 := Group( (1,2,3), (1,2) );
C3 := Group( (1,2,3) );
C2 := Group( (1,2) );
iso := rec( 1 := C3, 5 := S3,
6 := C2, 7 := C2 );
mu := [
[ [2], [1,2], [1,2,3], [1,2,4], x -> (1,2,3) ],
[ [2], [1,2], [1,2,4], [1,2,3], x -> (1,3,2) ]
];
dim := 3;
# 1: 9 x 72 matrix with rank 8 and kernel dimension 1. Time: 0.000 sec.
# 2: 72 x 261 matrix with rank 63 and kernel dimension 9. Time: 0.008 sec.
# 3: 261 x 1033 matrix with rank 197 and kernel dimension 64. Time: 0.440 sec.
# 4: 1033 x 5037 matrix with rank 834 and kernel dimension 199. Time: 33.354 sec.
# 5: 5037 x 26911 matrix with rank 4201 and kernel dimension 836. Time: 7623.669 sec.
# Cohomology dimension at degree 0: GF(3)^(1 x 1)
# Cohomology dimension at degree 1: GF(3)^(1 x 1)
# Cohomology dimension at degree 2: GF(3)^(1 x 1)
# Cohomology dimension at degree 3: GF(3)^(1 x 2)
# Cohomology dimension at degree 4: GF(3)^(1 x 2)
#-------------------------------------------------------------------------------
--------
#------------------------- new triangulation: -------------------------------
# 1: 9 x 105 matrix with rank 8 and kernel dimension 1. Time: 0.000 sec.
# 2: 105 x 736 matrix with rank 96 and kernel dimension 9. Time: 0.004 sec.
# 3: 736 x 5963 matrix with rank 638 and kernel dimension 98. Time: 0.196 sec.
# 4: 5963 x 53053 matrix with rank 5323 and kernel dimension 640. Time: 13.921 sec.
# 5: 53053 x 497007 matrix with rank 47728 and kernel dimension 5325. Time: 1054.590 sec.
# Cohomology dimension at degree 0: GF(2)^(1 x 1)
# Cohomology dimension at degree 1: GF(2)^(1 x 1)
# Cohomology dimension at degree 2: GF(2)^(1 x 2)
# Cohomology dimension at degree 3: GF(2)^(1 x 2)
# Cohomology dimension at degree 4: GF(2)^(1 x 2)
#----------------------------------------------->>>> Z^(1 x 1)
#----------------------------------------------->>>> 0
#----------------------------------------------->>>> Z/< 6 >
#----------------------------------------------->>>> Z/< 2 >
# in 3 h, 14 GB
#------------------------- old triangulation: --------------------------------
# matrix sizes:
# [ 9, 153, 2432, 50651, 1133693 ]
# factors:
# [ 17, 15.8954, 20.8269, 22.3824 ]
#cohomology over Z:
#------->>>> Z^(1 x 1)
#------->>>> 0
#------->>>> Z/< 6 >
#homology over Z:
#------->>>> Z^(1 x 1)
#------->>>> Z/< 6 >
#------->>>> Z/< 2 >
# 1: 9 x 153 matrix with rank 8 and kernel dimension 1. Time: 0.000 sec.
# 2: 153 x 2432 matrix with rank 144 and kernel dimension 9. Time: 0.052 sec.
# 3: 2432 x 50651 matrix with rank 2286 and kernel dimension 146. Time: 2.612 sec.
# 4: 50651 x 1133693 matrix with rank 48363 and kernel dimension 2288. Time: 1139.408 sec.
# 5: 1133693 x 25853535 exceeded memory (-o 15g)
# Cohomology dimension at degree 0: GF(2)^(1 x 1)
# Cohomology dimension at degree 1: GF(2)^(1 x 1)
# Cohomology dimension at degree 2: GF(2)^(1 x 2)
# Cohomology dimension at degree 3: GF(2)^(1 x 2)
