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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>

I hope this help you

Mario Pineda

Departamento deMatematicas

Universidad Autonoma Metropolitana-I

Mexico

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