How many homomorphisms are there from $C_2\times C_3$ to $S_4$ are there? (Using kernel and image to describe).
My thoughts/attempt: Determine homomorphisms by the image of the domain's generators. We know that the order of a map $\varphi(x)$ divides the order of $x$. We take every element in the first $C_2 \times C_3$ and map it to something. I'm guessing since 2 and 3 are relatively prime, $C_2\times C_3\cong C_6$ and we map every element of the cyclic group of order 6 to elements of the same order in $S_4$. But not sure where to go from here!
There is no element of order $6$ in $S_4$. You can verify this by thinking about the cycle structure of the elements of $S_4$.
Therefore, you may as well just look for non-trivial homomorphisms from $C_2$ to $S_4$, and from $C_3$ to $S_4$.
For $C_2=\{1,x\}$, the generator $x$ has to be sent to an element of order $2$, and there are $9$ of them.
For $C_3=\{1,y,y^2\}$, the generator $y$ has to be sent to a $3$-cycle. The number of $3$-cycles is ${4\choose 3}\times 2=8$.
Then there is the trivial homomorphism.
So $9+8+1=18$.