Simplest (smallest element) of non-cyclic abelian group I am looking for some examples of non-cyclic abelian groups. 
I found something like order 12, or other group. 
Here i am looking for simplest (smallest element) non-cylcic abelian groups. 
If you know something about this please let me know. 

The Klein-four-group is an abelian group with 4 elements, $1, a, b, c: a^2 = b^2 = c^2 = 1$ and, $ab = c, ac = b, bc = a$.
Other small non-abelian groups include $D_n$, the dihedral group, and $S_n$, the symmetric group. The order of $D_n$ is $2n$ and $S_n$ is $n!$.
Edit: I misread non-cyclic abelian groups as non-abelian. The family of dihedral groups is a product of abelian groups $Z_2, Z_n$ but the product is not direct (direct product would be abelian), it is a semi-direct product. 
