How many homomorphisms are there from $D_6$ to $D_5$? I know that:
$D_6$={$e,a,a^{2},a^{3},a^{4},a^{5},b,ab,a^{2}b,a^{3}b,a^{4}b,a^{5}b$}, with $a^{6}=e$ and $ba^{k}b=a^{-k}$.
$D_5$={$e,r,r^{2},r^{3},r^{4},s,rs,r^{2}s,r^{3}s,r^{4}s$}, with $r^{5}=e$ and $sr^{k}s=r^{-k}$.
Let $\varphi:D_6\rightarrow D_5$, a homomorphism is an injective function such that: $\forall x,y\in D_6:\varphi(xy)=\varphi(x)\varphi(y)$
I am now asked to determine how many homomorphisms there are from $D_6$ to $D_5$ but the only one i can find is the trivial one thus: $\varphi(e)=\varphi(a)=\varphi(a^{2})=\varphi(a^{3})=\varphi(a^{4})=\varphi(a^{5})=\varphi(b)=\varphi(ab)=\varphi(a^{2}b)=\varphi(a^{3}b)=\varphi(a^{4}b)=\varphi(a^{5}b)=r^{5}$
Can anyone help me with this i'm sure there have to be more homomorphism but i can't find them.

Hint: In light of the first isomorphism theorem, you want to find normal subgroups $N\triangleleft D_6$ and then find subgroups of $D_5$ isomorphic to $D_6/N$.
