174种1500阶群

gap> NumberSmallGroups(1500);
174
gap> L:=Factors(1500);
[ 2, 2, 3, 5, 5, 5 ]
gap> G:=AbelianGroup(L);;IdGroup(G);AbelianInvariants(G);
[ 1500, 174 ]
[ 2, 2, 3, 5, 5, 5 ]
gap> L1:=[L[1]*L[2],L[3],L[4],L[5],L[6]];
[ 4, 3, 5, 5, 5 ]
gap> G:=AbelianGroup(L1);;IdGroup(G);AbelianInvariants(G);
[ 1500, 111 ]
[ 3, 4, 5, 5, 5 ]
gap> L2:=[L[1]*L[2],L[3],L[4]*L[5],L[6]];
[ 4, 3, 25, 5 ]
gap> G:=AbelianGroup(L2);;IdGroup(G);AbelianInvariants(G);
[ 1500, 31 ]
[ 3, 4, 5, 25 ]
gap> L3:=[L[1]*L[2],L[3],L[4]*L[5]*L[6]];
[ 4, 3, 125 ]
gap> G:=AbelianGroup(L3);;IdGroup(G);AbelianInvariants(G);
[ 1500, 4 ]
[ 3, 4, 125 ]
gap> G:=CyclicGroup(1500);;IdGroup(G);AbelianInvariants(G);
[ 1500, 4 ]
[ 3, 4, 125 ]
gap> L4:=[L[1],L[2],L[3],L[4]*L[5]*L[6]];
[ 2, 2, 3, 125 ]
gap> G:=AbelianGroup(L4);;IdGroup(G);AbelianInvariants(G);
[ 1500, 12 ]
[ 2, 2, 3, 125 ]
gap> L5:=[L[1],L[2],L[3],L[4]*L[5],L[6]];
[ 2, 2, 3, 25, 5 ]
gap> G:=AbelianGroup(L5);;IdGroup(G);AbelianInvariants(G);
[ 1500, 95 ]
[ 2, 2, 3, 5, 25 ]
共有6种1500阶的交换群，不变因子分别为：
{5，5，60}，{5，300}，{1500}，{5，10，30}，{10，150}， {2，750}。