174種1500階群

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種1500階的交換群，不變因子分别為：

{5，5，60}，{5，300}，{1500}，{5，10，30}，{10，150}， {2，750}。