> < ^ Date: Wed, 15 Aug 2001 11:22:31 -0500
> < ^ From: Mario Pineda <mpr@xanum.uam.mx >
< ^ Subject: Re: LatticeSubgroups problem

Dear Gordon,

I calcutae the lattice of your group with GAP 3.4r4 and I obtain the following:

```gap> g := SmallGroup(144,32);
144_32
gap> l:=Lattice(g);
LatticeSubgroups( 144_32 )

gap> SetPrintLevel(l,1);
gap> l;
#I  class 1, size 1, length 1
#I  class 2, size 2, length 1
#I  class 3, size 2, length 36
#I  class 4, size 3, length 1
#I  class 5, size 4, length 3
#I  class 6, size 4, length 18
#I  class 7, size 6, length 1
#I  class 8, size 6, length 12
#I  class 9, size 8, length 1
#I  class 10, size 8, length 9
#I  class 11, size 8, length 9
#I  class 12, size 9, length 4
#I  class 13, size 12, length 3
#I  class 14, size 12, length 6
#I  class 15, size 16, length 9
#I  class 16, size 18, length 4
#I  class 17, size 18, length 4
#I  class 18, size 18, length 4
#I  class 19, size 24, length 1
#I  class 20, size 24, length 3
#I  class 21, size 24, length 3
#I  class 22, size 36, length 4
#I  class 23, size 48, length 3
#I  class 24, size 72, length 1
#I  class 25, size 144, length 1
LatticeSubgroups( 144_32 )
gap> SetPrintLevel(l,2);
gap> l;
#I  class 1, size 1, length 1
#I    representative [  ]
#I      maximals
#I  class 2, size 2, length 1
#I    representative [ f ]
#I      maximals [ 1, 1 ]
#I  class 3, size 2, length 36
#I    representative [ a ]
#I      maximals [ 1, 1 ]
#I  class 4, size 3, length 1
#I    representative [ c ]
#I      maximals [ 1, 1 ]
#I  class 5, size 4, length 3
#I    representative [ f, e ]
#I      maximals [ 2, 1 ]
#I  class 6, size 4, length 18
#I    representative [ f, a ]
#I      maximals [ 3, 1 ] [ 3, 4 ] [ 2, 1 ]
#I  class 7, size 6, length 1
#I    representative [ f, c ]
#I      maximals [ 4, 1 ] [ 2, 1 ]
#I  class 8, size 6, length 12
#I    representative [ c, a ]
#I      maximals [ 4, 1 ] [ 3, 1 ] [ 3, 5 ] [ 3, 9 ]
#I  class 9, size 8, length 1
#I    representative [ f, e, d*f ]
#I      maximals [ 5, 1 ] [ 5, 2 ] [ 5, 3 ]
#I  class 10, size 8, length 9
#I    representative [ f, e, a*b ]
#I      maximals [ 6, 17 ] [ 6, 18 ] [ 5, 1 ]
#I  class 11, size 8, length 9
#I    representative [ f, e, a*b*d*f ]
#I      maximals [ 5, 1 ]
#I  class 12, size 9, length 4
#I    representative [ c, b ]
#I      maximals [ 4, 1 ]
#I  class 13, size 12, length 3
#I    representative [ f, e, c ]
#I      maximals [ 7, 1 ] [ 5, 1 ]
#I  class 14, size 12, length 6
#I    representative [ f, c, a ]
#I      maximals [ 8, 1 ] [ 8, 4 ] [ 7, 1 ] [ 6, 1 ] [ 6, 3 ] [ 6, 5 ]
#I  class 15, size 16, length 9
#I    representative [ f, e, d*f, a ]
#I      maximals [ 11, 6 ] [ 10, 6 ] [ 9, 1 ]
#I  class 16, size 18, length 4
#I    representative [ f, c, b ]
#I      maximals [ 12, 1 ] [ 7, 1 ]
#I  class 17, size 18, length 4
#I    representative [ c, b, a ]
#I      maximals [ 12, 1 ] [ 8, 1 ] [ 8, 5 ] [ 8, 9 ]
#I  class 18, size 18, length 4
#I    representative [ c, b, a*f ]
#I      maximals [ 12, 1 ] [ 8, 4 ] [ 8, 7 ] [ 8, 10 ]
#I  class 19, size 24, length 1
#I    representative [ f, e, d*f, c ]
#I      maximals [ 13, 1 ] [ 13, 2 ] [ 13, 3 ] [ 9, 1 ]
#I  class 20, size 24, length 3
#I    representative [ f, e, c, a*b ]
#I      maximals [ 14, 5 ] [ 14, 6 ] [ 13, 1 ] [ 10, 1 ] [ 10, 2 ] [ 10, 3 ]
#I  class 21, size 24, length 3
#I    representative [ f, e, c, a*b*d*f ]
#I      maximals [ 13, 1 ] [ 11, 1 ] [ 11, 2 ] [ 11, 3 ]
#I  class 22, size 36, length 4
#I    representative [ f, c, b, a ]
#I      maximals [ 18, 1 ] [ 17, 1 ] [ 16, 1 ] [ 14, 1 ] [ 14, 3 ] [ 14, 5 ]
#I  class 23, size 48, length 3
#I    representative [ f, e, d*f, c, a ]
#I      maximals [ 21, 2 ] [ 20, 2 ] [ 19, 1 ] [ 15, 1 ] [ 15, 2 ] [ 15, 3 ]
#I  class 24, size 72, length 1
#I    representative [ f, e, d*f, c, b ]
#I      maximals [ 19, 1 ] [ 16, 1 ] [ 16, 2 ] [ 16, 3 ] [ 16, 4 ]
#I  class 25, size 144, length 1
#I    representative [ a, b, c, d, e, f ]
#I      maximals [ 24, 1 ] [ 23, 1 ] [ 23, 2 ] [ 23, 3 ] [ 22, 1 ] [ 22, 2 ]
[ 22, 3 ] [ 22, 4 ]
LatticeSubgroups( 144_32 )
gap> SetPrintLevel(l,3);
gap> l;
#I  class 1, size 1, length 1
#I    representative [  ]
#I      maximals
#I  class 2, size 2, length 1
#I    representative [ f ]
#I      maximals [ 1, 1 ]
#I  class 3, size 2, length 36
#I    representative [ a ]
#I      maximals [ 1, 1 ]
#I    conjugate 2 by e is [ a*d*e*f ]
#I    conjugate 3 by d is [ a*d*e ]
#I    conjugate 4 by d*e*f is [ a*f ]
#I    conjugate 5 by c is [ a*c^2 ]
#I    conjugate 6 by c*e is [ a*c^2*d*e*f ]
#I    conjugate 7 by c*d is [ a*c^2*d*e ]
#I    conjugate 8 by c*d*e*f is [ a*c^2*f ]
#I    conjugate 9 by c^2 is [ a*c ]
#I    conjugate 10 by c^2*e is [ a*c*d*e*f ]
#I    conjugate 11 by c^2*d is [ a*c*d*e ]
#I    conjugate 12 by c^2*d*e*f is [ a*c*f ]
#I    conjugate 13 by b is [ a*b^2 ]
#I    conjugate 14 by b*e*f is [ a*b^2*d*f ]
#I    conjugate 15 by b*d is [ a*b^2*f ]
#I    conjugate 16 by b*d*e is [ a*b^2*d ]
#I    conjugate 17 by b*c is [ a*b^2*c^2 ]
#I    conjugate 18 by b*c*e*f is [ a*b^2*c^2*d*f ]
#I    conjugate 19 by b*c*d is [ a*b^2*c^2*f ]
#I    conjugate 20 by b*c*d*e is [ a*b^2*c^2*d ]
#I    conjugate 21 by b*c^2 is [ a*b^2*c ]
#I    conjugate 22 by b*c^2*e*f is [ a*b^2*c*d*f ]
#I    conjugate 23 by b*c^2*d is [ a*b^2*c*f ]
#I    conjugate 24 by b*c^2*d*e is [ a*b^2*c*d ]
#I    conjugate 25 by b^2 is [ a*b*c^2 ]
#I    conjugate 26 by b^2*e*f is [ a*b*c^2*f ]
#I    conjugate 27 by b^2*d*f is [ a*b*c^2*e*f ]
#I    conjugate 28 by b^2*d*e*f is [ a*b*c^2*e ]
#I    conjugate 29 by b^2*c is [ a*b*c ]
#I    conjugate 30 by b^2*c*e*f is [ a*b*c*f ]
#I    conjugate 31 by b^2*c*d*f is [ a*b*c*e*f ]
#I    conjugate 32 by b^2*c*d*e*f is [ a*b*c*e ]
#I    conjugate 33 by b^2*c^2 is [ a*b ]
#I    conjugate 34 by b^2*c^2*e*f is [ a*b*f ]
#I    conjugate 35 by b^2*c^2*d*f is [ a*b*e*f ]
#I    conjugate 36 by b^2*c^2*d*e*f is [ a*b*e ]
#I  class 4, size 3, length 1
#I    representative [ c ]
#I      maximals [ 1, 1 ]
#I  class 5, size 4, length 3
#I    representative [ f, e ]
#I      maximals [ 2, 1 ]
#I    conjugate 2 by b is [ f, d*e ]
#I    conjugate 3 by b^2 is [ f, d*f ]
#I  class 6, size 4, length 18
#I    representative [ f, a ]
#I      maximals [ 3, 1 ] [ 3, 4 ] [ 2, 1 ]
#I    conjugate 2 by e is [ f, a*d*e*f ]
#I    conjugate 3 by c is [ f, a*c^2 ]
#I    conjugate 4 by c*e is [ f, a*c^2*d*e*f ]
#I    conjugate 5 by c^2 is [ f, a*c ]
#I    conjugate 6 by c^2*e is [ f, a*c*d*e*f ]
#I    conjugate 7 by b is [ f, a*b^2 ]
#I    conjugate 8 by b*d*e is [ f, a*b^2*d ]
#I    conjugate 9 by b*c is [ f, a*b^2*c^2 ]
#I    conjugate 10 by b*c*d*e is [ f, a*b^2*c^2*d ]
#I    conjugate 11 by b*c^2 is [ f, a*b^2*c ]
#I    conjugate 12 by b*c^2*d*e is [ f, a*b^2*c*d ]
#I    conjugate 13 by b^2 is [ f, a*b*c^2 ]
#I    conjugate 14 by b^2*d*f is [ f, a*b*c^2*e*f ]
#I    conjugate 15 by b^2*c is [ f, a*b*c ]
#I    conjugate 16 by b^2*c*d*f is [ f, a*b*c*e*f ]
#I    conjugate 17 by b^2*c^2 is [ f, a*b ]
#I    conjugate 18 by b^2*c^2*d*f is [ f, a*b*e*f ]
#I  class 7, size 6, length 1
#I    representative [ f, c ]
#I      maximals [ 4, 1 ] [ 2, 1 ]
#I  class 8, size 6, length 12
#I    representative [ c, a ]
#I      maximals [ 4, 1 ] [ 3, 1 ] [ 3, 5 ] [ 3, 9 ]
#I    conjugate 2 by e is [ c, a*d*e*f ]
#I    conjugate 3 by d is [ c, a*d*e ]
#I    conjugate 4 by d*e*f is [ c, a*f ]
#I    conjugate 5 by b is [ c, a*b^2 ]
#I    conjugate 6 by b*e*f is [ c, a*b^2*d*f ]
#I    conjugate 7 by b*d is [ c, a*b^2*f ]
#I    conjugate 8 by b*d*e is [ c, a*b^2*d ]
#I    conjugate 9 by b^2 is [ c, a*b*c^2 ]
#I    conjugate 10 by b^2*e*f is [ c, a*b*c^2*f ]
#I    conjugate 11 by b^2*d*f is [ c, a*b*c^2*e*f ]
#I    conjugate 12 by b^2*d*e*f is [ c, a*b*c^2*e ]
#I  class 9, size 8, length 1
#I    representative [ f, e, d*f ]
#I      maximals [ 5, 1 ] [ 5, 2 ] [ 5, 3 ]
#I  class 10, size 8, length 9
#I    representative [ f, e, a*b ]
#I      maximals [ 6, 17 ] [ 6, 18 ] [ 5, 1 ]
#I    conjugate 2 by c is [ f, e, a*b*c^2 ]
#I    conjugate 3 by c^2 is [ f, e, a*b*c ]
#I    conjugate 4 by b is [ f, d*e, a*c^2 ]
#I    conjugate 5 by b*c is [ f, d*e, a*c ]
#I    conjugate 6 by b*c^2 is [ f, d*e, a ]
#I    conjugate 7 by b^2 is [ f, d*f, a*b^2*c^2 ]
#I    conjugate 8 by b^2*c is [ f, d*f, a*b^2*c ]
#I    conjugate 9 by b^2*c^2 is [ f, d*f, a*b^2 ]
#I  class 11, size 8, length 9
#I    representative [ f, e, a*b*d*f ]
#I      maximals [ 5, 1 ]
#I    conjugate 2 by c is [ f, e, a*b*c^2*d*f ]
#I    conjugate 3 by c^2 is [ f, e, a*b*c*d*f ]
#I    conjugate 4 by b is [ f, d*e, a*c^2*e ]
#I    conjugate 5 by b*c is [ f, d*e, a*c*e ]
#I    conjugate 6 by b*c^2 is [ f, d*e, a*e ]
#I    conjugate 7 by b^2 is [ f, d*f, a*b^2*c^2*d*e ]
#I    conjugate 8 by b^2*c is [ f, d*f, a*b^2*c*d*e ]
#I    conjugate 9 by b^2*c^2 is [ f, d*f, a*b^2*d*e ]
#I  class 12, size 9, length 4
#I    representative [ c, b ]
#I      maximals [ 4, 1 ]
#I    conjugate 2 by e is [ c, b*d ]
#I    conjugate 3 by d is [ c, b*d*e*f ]
#I    conjugate 4 by d*e*f is [ c, b*e*f ]
#I  class 13, size 12, length 3
#I    representative [ f, e, c ]
#I      maximals [ 7, 1 ] [ 5, 1 ]
#I    conjugate 2 by b is [ f, d*e, c ]
#I    conjugate 3 by b^2 is [ f, d*f, c ]
#I  class 14, size 12, length 6
#I    representative [ f, c, a ]
#I      maximals [ 8, 1 ] [ 8, 4 ] [ 7, 1 ] [ 6, 1 ] [ 6, 3 ] [ 6, 5 ]
#I    conjugate 2 by e is [ f, c, a*d*e*f ]
#I    conjugate 3 by b is [ f, c, a*b^2 ]
#I    conjugate 4 by b*d*e is [ f, c, a*b^2*d ]
#I    conjugate 5 by b^2 is [ f, c, a*b*c^2 ]
#I    conjugate 6 by b^2*d*f is [ f, c, a*b*c^2*e*f ]
#I  class 15, size 16, length 9
#I    representative [ f, e, d*f, a ]
#I      maximals [ 11, 6 ] [ 10, 6 ] [ 9, 1 ]
#I    conjugate 2 by c is [ f, e, d*f, a*c^2 ]
#I    conjugate 3 by c^2 is [ f, e, d*f, a*c ]
#I    conjugate 4 by b is [ f, d*e, e, a*b^2 ]
#I    conjugate 5 by b*c is [ f, d*e, e, a*b^2*c^2 ]
#I    conjugate 6 by b*c^2 is [ f, d*e, e, a*b^2*c ]
#I    conjugate 7 by b^2 is [ f, d*f, d*e, a*b*c^2 ]
#I    conjugate 8 by b^2*c is [ f, d*f, d*e, a*b*c ]
#I    conjugate 9 by b^2*c^2 is [ f, d*f, d*e, a*b ]
#I  class 16, size 18, length 4
#I    representative [ f, c, b ]
#I      maximals [ 12, 1 ] [ 7, 1 ]
#I    conjugate 2 by e is [ f, c, b*d ]
#I    conjugate 3 by d is [ f, c, b*d*e*f ]
#I    conjugate 4 by d*e*f is [ f, c, b*e*f ]
#I  class 17, size 18, length 4
#I    representative [ c, b, a ]
#I      maximals [ 12, 1 ] [ 8, 1 ] [ 8, 5 ] [ 8, 9 ]
#I    conjugate 2 by e is [ c, b*d, a*d*e*f ]
#I    conjugate 3 by d is [ c, b*d*e*f, a*d*e ]
#I    conjugate 4 by d*e*f is [ c, b*e*f, a*f ]
#I  class 18, size 18, length 4
#I    representative [ c, b, a*f ]
#I      maximals [ 12, 1 ] [ 8, 4 ] [ 8, 7 ] [ 8, 10 ]
#I    conjugate 2 by e is [ c, b*d, a*d*e ]
#I    conjugate 3 by d is [ c, b*d*e*f, a*d*e*f ]
#I    conjugate 4 by d*e*f is [ c, b*e*f, a ]
#I  class 19, size 24, length 1
#I    representative [ f, e, d*f, c ]
#I      maximals [ 13, 1 ] [ 13, 2 ] [ 13, 3 ] [ 9, 1 ]
#I  class 20, size 24, length 3
#I    representative [ f, e, c, a*b ]
#I      maximals [ 14, 5 ] [ 14, 6 ] [ 13, 1 ] [ 10, 1 ] [ 10, 2 ] [ 10, 3 ]
#I    conjugate 2 by b is [ f, d*e, c, a*c^2 ]
#I    conjugate 3 by b^2 is [ f, d*f, c, a*b^2*c^2 ]
#I  class 21, size 24, length 3
#I    representative [ f, e, c, a*b*d*f ]
#I      maximals [ 13, 1 ] [ 11, 1 ] [ 11, 2 ] [ 11, 3 ]
#I    conjugate 2 by b is [ f, d*e, c, a*c^2*e ]
#I    conjugate 3 by b^2 is [ f, d*f, c, a*b^2*c^2*d*e ]
#I  class 22, size 36, length 4
#I    representative [ f, c, b, a ]
#I      maximals [ 18, 1 ] [ 17, 1 ] [ 16, 1 ] [ 14, 1 ] [ 14, 3 ] [ 14, 5 ]
#I    conjugate 2 by e is [ f, c, b*d, a*d*e*f ]
#I    conjugate 3 by d is [ f, c, b*d*e*f, a*d*e ]
#I    conjugate 4 by d*e*f is [ f, c, b*e*f, a*f ]
#I  class 23, size 48, length 3
#I    representative [ f, e, d*f, c, a ]
#I      maximals [ 21, 2 ] [ 20, 2 ] [ 19, 1 ] [ 15, 1 ] [ 15, 2 ] [ 15, 3 ]
#I    conjugate 2 by b is [ f, d*e, e, c, a*b^2 ]
#I    conjugate 3 by b^2 is [ f, d*f, d*e, c, a*b*c^2 ]
#I  class 24, size 72, length 1
#I    representative [ f, e, d*f, c, b ]
#I      maximals [ 19, 1 ] [ 16, 1 ] [ 16, 2 ] [ 16, 3 ] [ 16, 4 ]
#I  class 25, size 144, length 1
#I    representative [ a, b, c, d, e, f ]
#I      maximals [ 24, 1 ] [ 23, 1 ] [ 23, 2 ] [ 23, 3 ] [ 22, 1 ] [ 22, 2 ]
[ 22, 3 ] [ 22, 4 ]
LatticeSubgroups( 144_32 )
gap>
```