|
|||
|---|---|---|---|
| The
following commands compute Poincare functions for most of the groups of
order 64. A few groups are omitted from this general calculation as
they need too much RAM. Again, we
have no general proof that our algorithm produces a series which is
correct in
arbitrarily high degree. Proofs must be given case by case. |
|||
| gap>
L:=[60,67,73,80,88,99,199,206,213,215,216,224,227,232,241,242,266];; gap> for G in AllSmallGroups(64) do > if not IdSmallGroup(G)[2] in L then > Print("\n Small group ",IdSmallGroup(G), " has Poincare series \n", > PoincareSeries(G), "\n"); > fi; > od; Small group [ 64, 1 ] has Poincare series (1)/(-x+1) Small group [ 64, 2 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 3 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 4 ] has Poincare series (x^2-x+1)/(x^7-3*x^6+3*x^5-x^4-x^3+3*x^2-3*x+1) Small group [ 64, 5 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 6 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 7 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 8 ] has Poincare series (x^4+x^3+x^2+x+1)/(-x^7+x^6+x^5-x^4+x^3-x^2-x+1) Small group [ 64, 9 ] has Poincare series (2*x^4+x^3+x^2+x+1)/(-x^9+x^8+2*x^5-2*x^4-x+1) Small group [ 64, 10 ] has Poincare series (1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 11 ] has Poincare series (x^4+x^2+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 12 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 13 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 14 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 15 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 16 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 17 ] has Poincare series (x^2+x+1)/(-x^5+x^4+2*x^3-2*x^2-x+1) Small group [ 64, 18 ] has Poincare series (-x^6+x^5+x^4-2*x^3+2*x^2-x+1)/(-x^9+3*x^8-4*x^7+ 4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 19 ] has Poincare series (-x^6+x^5+x^4-2*x^3+2*x^2-x+1)/(-x^9+3*x^8-4*x^7+ 4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 20 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 21 ] has Poincare series (x^2+x+1)/(-x^5+x^4+2*x^3-2*x^2-x+1) Small group [ 64, 22 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 23 ] has Poincare series (-x^3+2*x^2-x+1)/(x^7-3*x^6+3*x^5-x^4-x^3+3*x^2-3*x+1) Small group [ 64, 24 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 25 ] has Poincare series (-x^6+x^5+x^4-2*x^3+2*x^2-x+1)/(-x^9+3*x^8-4*x^7+ 4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 26 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 27 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 28 ] has Poincare series (1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 29 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 30 ] has Poincare series (x^4-x^3+x^2-x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 31 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 32 ] has Poincare series (x^2-x+1)/(x^7-3*x^6+3*x^5-x^4-x^3+3*x^2-3*x+1) Small group [ 64, 33 ] has Poincare series (x^6+x^3+1)/(-x^10+2*x^9-x^8+x^2-2*x+1) Small group [ 64, 34 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 35 ] has Poincare series (-x^7+x^6+x^5-x^3+x^2+1)/(-x^10+2*x^9-x^8+x^2-2*x+1) Small group [ 64, 36 ] has Poincare series (x^5-x^4+x^2-x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 37 ] has Poincare series (x^5+x^2+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 38 ] has Poincare series (1)/(-x^4+2*x^3-2*x+1) Small group [ 64, 39 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 40 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 41 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 42 ] has Poincare series (x^2-x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 43 ] has Poincare series (x^5+x^2+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 44 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 45 ] has Poincare series (1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 46 ] has Poincare series (1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 47 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 48 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 49 ] has Poincare series (1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 50 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 51 ] has Poincare series (1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 52 ] has Poincare series (1)/(x^2-2*x+1) Small group [ 64, 53 ] has Poincare series (1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 54 ] has Poincare series (x^2+x+1)/(-x^3+x^2-x+1) Small group [ 64, 55 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 56 ] has Poincare series (x^2+x+1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x+1) Small group [ 64, 57 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 58 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 59 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 61 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 62 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 63 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 64 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 65 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 66 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 68 ] has Poincare series (x^4+2*x^3+x^2+2*x+1)/(-x^7+x^6+x^5-x^4+x^3-x^2-x+1) Small group [ 64, 69 ] has Poincare series (-x^2+x+1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x+1) Small group [ 64, 70 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 71 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 72 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 74 ] has Poincare series (-x^3+x^2+1)/(x^7-3*x^6+3*x^5-x^4-x^3+3*x^2-3*x+1) Small group [ 64, 75 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 76 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 77 ] has Poincare series (-x^2+x+1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x+1) Small group [ 64, 78 ] has Poincare series (-x^2+x+1)/(x^6-2*x^5-x^4+4*x^3-x^2-2*x+1) Small group [ 64, 79 ] has Poincare series (x^4+2*x^3+x^2+2*x+1)/(-x^7+x^6+x^5-x^4+x^3-x^2-x+1) Small group [ 64, 81 ] has Poincare series (x^4+2*x^3+x^2+2*x+1)/(-x^7+x^6+x^5-x^4+x^3-x^2-x+1) Small group [ 64, 82 ] has Poincare series (x^8+2*x^7+3*x^6+6*x^5+6*x^4+6*x^3+3*x^2+2*x+1)/(-x^11+ x^10-x^9+x^8+2*x^7-2*x^\6+2*x^5-2*x^4-x^3+x^2-x+1) Small group [ 64, 83 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 84 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 85 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 86 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 87 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 89 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 90 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 91 ] has Poincare series (x^5+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 92 ] has Poincare series (x^2-x+1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 93 ] has Poincare series (x^5+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 94 ] has Poincare series (x^5+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 95 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 96 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 97 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 98 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 100 ] has Poincare series (x^5+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 101 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 102 ] has Poincare series (x^5+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 103 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 104 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 105 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 106 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 107 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 108 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 109 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 110 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 111 ] has Poincare series (-x^6+x^5+x^4-x^3+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 112 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 113 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 114 ] has Poincare series (-x^4+x^3+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 115 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 116 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 117 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 118 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 119 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 120 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 121 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 122 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 123 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 124 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 125 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 126 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 127 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 128 ] has Poincare series (1)/(x^5-3*x^4+2*x^3+2*x^2-3*x+1) Small group [ 64, 129 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 130 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 131 ] has Poincare series (x^2-x+1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 132 ] has Poincare series (x^2+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 133 ] has Poincare series (x^2+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 134 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 135 ] has Poincare series (-x^6+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 136 ] has Poincare series (x^5+x^4-x^3+x^2+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3 +4*x^2-3*x+1) Small group [ 64, 137 ] has Poincare series (x^6+x^5+2*x^4+x^3+2*x^2+x+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4- 2*x^3+2*x^2-2*x+1) Small group [ 64, 138 ] has Poincare series (-x^3+x^2+1)/(x^7-3*x^6+3*x^5-x^4-x^3+3*x^2-3*x+1) Small group [ 64, 139 ] has Poincare series (x^7+2*x^5+x^3+x^2+x+1)/(-x^10+2*x^9-x^8+x^2-2*x+1) Small group [ 64, 140 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 141 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 142 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 143 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 144 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 145 ] has Poincare series (-x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 146 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 147 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 148 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 149 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 150 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 151 ] has Poincare series (-x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 152 ] has Poincare series (-x^6+x^5+x^4-x^3+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3 +4*x^2-3*x+1) Small group [ 64, 153 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 154 ] has Poincare series (x^6+x^5+x^2+x+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 155 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 156 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 157 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 158 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 159 ] has Poincare series (2*x^3+x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 160 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 161 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 162 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 163 ] has Poincare series (2*x^3+x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 164 ] has Poincare series (2*x^3+x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 165 ] has Poincare series (2*x^3+x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 166 ] has Poincare series (-x^4+x^3+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 167 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 168 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 169 ] has Poincare series (-x^4+x^3+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 170 ] has Poincare series (-x^4+x^3+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 171 ] has Poincare series (2*x^3+x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 172 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 173 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 174 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 175 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 176 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 177 ] has Poincare series (x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 178 ] has Poincare series (-x^3+x+1)/(-x^6+2*x^5-x^4+x^2-2*x+1) Small group [ 64, 179 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 180 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 181 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 182 ] has Poincare series (x^4+x^3+x^2+x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 183 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 184 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 185 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 186 ] has Poincare series (1)/(-x^3+3*x^2-3*x+1) Small group [ 64, 187 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 188 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 189 ] has Poincare series (x^2+x+1)/(x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 190 ] has Poincare series (1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 191 ] has Poincare series (x^6+x^5+x^2+x+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 192 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 193 ] has Poincare series (1)/(-x^6+4*x^5-5*x^4+5*x^2-4*x+1) Small group [ 64, 194 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 195 ] has Poincare series (1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 196 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 197 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 198 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 200 ] has Poincare series (x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 201 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 202 ] has Poincare series (1)/(-x^6+4*x^5-5*x^4+5*x^2-4*x+1) Small group [ 64, 203 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 204 ] has Poincare series (1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 205 ] has Poincare series (x^3+x+1)/(x^7-3*x^6+3*x^5-x^4-x^3+3*x^2-3*x+1) Small group [ 64, 207 ] has Poincare series (1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 208 ] has Poincare series (x^4+x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 209 ] has Poincare series (-x^4+x^3+1)/(x^8-4*x^7+8*x^6-12*x^5+14*x^4-12*x^3+8*x^2-4*x+1) Small group [ 64, 210 ] has Poincare series (x^4+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 211 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 212 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 214 ] has Poincare series (-x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 217 ] has Poincare series (x^6+x^5+x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 218 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 217 ] has Poincare series (x^6+x^5+x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 218 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 220 ] has Poincare series (x^5+2*x^4+2*x^2+2*x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 221 ] has Poincare series (x^4+x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 222 ] has Poincare series (-x^4-x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 223 ] has Poincare series (x^4-x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 225 ] has Poincare series (-x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 226 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 228 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 229 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 230 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 231 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 233 ] has Poincare series (x^4-x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 234 ] has Poincare series (x^4+x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 235 ] has Poincare series (x^4+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 236 ] has Poincare series (x^4+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 237 ] has Poincare series (-x^3+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 238 ] has Poincare series (x^4+2*x^3+3*x^2+2*x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 239 ] has Poincare series (x^4+2*x^3+3*x^2+2*x+1)/(x^6-2*x^5+3*x^4-4*x^3+3*x^2-2*x+1) Small group [ 64, 240 ] has Poincare series (x^4+x^2+x+1)/(-x^7+3*x^6-5*x^5+7*x^4-7*x^3+5*x^2-3*x+1) Small group [ 64, 243 ] has Poincare series (x^5+2*x^4+2*x^2+2*x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 244 ] has Poincare series (x^5-2*x^4+2*x^2+2*x+1)/(-x^8+2*x^7-2*x^6+2*x^5-2*x^3+2*x^2-2*x+1) Small group [ 64, 245 ] has Poincare series (x^12+2*x^11+3*x^10+2*x^9+5*x^8+x^7+9*x^6+x^5+5*x^4+ 2*x^3+3*x^2+2*x+1)/(x^14-2*x^13+3*x^12-4*x^11+5*x^10-6*x^9+ 7*x^8-8*x^7+7*x^6-6*x^5+5*x^4-4*x^3+3*x^2-2*x\+1) Small group [ 64, 246 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 247 ] has Poincare series (1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 248 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 249 ] has Poincare series (x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 250 ] has Poincare series (1)/(x^4-4*x^3+6*x^2-4*x+1) Small group [ 64, 251 ] has Poincare series (1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 252 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 253 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 254 ] has Poincare series (1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 255 ] has Poincare series (x^6+x^5+x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 256 ] has Poincare series (x^6+x^5+x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 257 ] has Poincare series (x^2+x+1)/(-x^5+3*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 258 ] has Poincare series (x^2+x+1)/(-x^9+3*x^8-4*x^7+4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 259 ] has Poincare series (x^6+2*x^5+3*x^4+3*x^3+3*x^2+2*x+1)/(x^8-2*x^7+2*x^6-2*x^5+ 2*x^4-2*x^3+2*x^2-2*x+1) Small group [ 64, 260 ] has Poincare series (1)/(-x^5+5*x^4-10*x^3+10*x^2-5*x+1) Small group [ 64, 261 ] has Poincare series (1)/(-x^5+5*x^4-10*x^3+10*x^2-5*x+1) Small group [ 64, 262 ] has Poincare series (x^2+x+1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 263 ] has Poincare series (x^2+x+1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 264 ] has Poincare series (x^2+x+1)/(x^6-4*x^5+7*x^4-8*x^3+7*x^2-4*x+1) Small group [ 64, 265 ] has Poincare series (x^6+2*x^5+3*x^4+3*x^3+3*x^2+2*x+1)/(-x^9+3*x^8-4*x^7+ 4*x^6-4*x^5+4*x^4-4*x^3+4*x^2-3*x+1) Small group [ 64, 267 ] has Poincare series (1)/(x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1) |
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