Show that $D_{11} \oplus \mathbb{Z}_3 \not \cong D_3 \oplus \mathbb{Z}_{11}$ 
Show that $D_{11} \oplus \mathbb{Z}_3 \not \cong D_3 \oplus \mathbb{Z}_{11}$

The first thing I check is that the size of both groups is the same since 
$|D_{11}|=22$ and $|\mathbb{Z}_3|=3$ thus $|D_{11} \oplus \mathbb{Z}_3=22\cdot 3=66$ and likewise $|D_3|=6$ and $|\mathbb{Z}_{11}|=11$ thus $|D_3 \oplus \mathbb{Z}_{11}|=6\cdot 11=66$
Could I evaluate the order of the centers of each group to show they are not the same size thus not isomoprhic? I see that the $|Z(D_{11} \oplus \mathbb{Z}_3)|\not = |Z(D_3 \oplus \mathbb{Z}_{11})|$ since the order of both center of the dihedral groups is only the identity where as the order of the center of $\mathbb{Z}_3$ is not the same size as the order of the center of $\mathbb{Z}_{11}$ 

As noted the center of both groups is different, hence the groups cannot be isomorphic. We have $Z(D_{2n+1})=1$ for all $n\ge 1$ and thus
$$
Z(D_{11}\times C_3)=Z(D_{11})\times Z(C_3)=1\times C_3\cong C_3, 
$$
and
$$
Z(D_3\times C_{11})=Z(D_3)\times Z(C_{11})=1\times C_{11}\cong C_{11}.
$$
Reference: Center of dihedral group
