List all the generators of the $\mathbb{Z}_{23} \times \mathbb{Z}_{12} /\left<(0,7)\right>$ 
List all the generators of the $\mathbb{Z}_{23} \times \mathbb{Z}_{12} /\left<(0,7)\right>$.

I know that a factor group $G/H$ is cyclic iff group $G$ is cyclic and $H$ its a subgroup of $G$. $\mathbb{Z}_{23} \times \mathbb{Z}_{12}$ is cyclic because $\gcd(23,12)=1$. $\mathbb{Z}_{23}$ have 22 elements of order 23 (all except 1) and $\mathbb{Z}_{12}$ have only 4 elements of order 12 $(1,5,7,11)$.
Then the generator of $\mathbb{Z}_{23} \times \mathbb{Z}_{12}$ could be $(a,1)$, $(a,5)$, $(a,7)$, and $(a,11)$ where $a$ is an element of $\mathbb{Z}_{23}$ different to $0$. $H$ have this generator $\left<(0,7)\right>$. How can I find the generator of $\mathbb{Z}_{23} \times \mathbb{Z}_{12} /\left<(0,7)\right>$?
 A: If $C$ is cyclic, then any generator of $C$ descends to a generator of $C/H$ in a quotient of $C$ (or more generally, any collection of generators of $G$ become a collection of generators for $G/H$, although in general, a minimal generating set need not be sent to a minimal generating set, e.g. $\mathbb Z \times \mathbb Z/\langle 0,1 \rangle \equiv \mathbb Z$).  Therefore, the answer is $(a,b)$ where $a\neq 0$ and $b\in \{1,5,7,11\}$.
However, I suspect that your confusion lies in the fact that listing the generators this way doesn't tell you when two generators in $G$ map to the same generator of $G/H$, so that your list of generators of $G/H$ contains duplicates, in a sense.  To resolve this, show that $H=\{0\}\times \mathbb Z_{12}$, so that $(a,b)$ is represented by $(a,0)$, so a minimal collection of possible generators is $(a,0)$ with $a\neq 0$.
A: I wouldn't identify $G = \mathbb Z_{23} \times \mathbb Z_{17}$ with a cyclic group first. First, I'd find a way to more explicitly identify the subgroup $H = \langle(0,7)\rangle$ and the quotient. 
Once you find out what this $H$ is (as in, a list of all its elements), I believe you will have no trouble identifying $G/H$ up to isomorphism, and its order (number of elements) should give you the generators, based on what you already know.
