TITLE: is the group cyclic, if Normal Subgroup and Quotient group are cyclic.
QUESTION [0 upvotes]: Prove or disprove that if $H$ is a normal subgroup of a group $G$ such that $H$ and $\frac{G}{H}$ are cyclic, then $G$ is cyclic. 
I am not sure if the above question has any mistake. 
I know examples where $\frac{G}{H}$ is cyclic, but $G$ is not cyclic. It is
$$G = S_3 = \{ I, (12),(23),(31), (123), (132) \}$$
and $$H = \{I,(123),(132)\}$$ H is a normal subgroup of G. 
$\frac{G}{H}$ is cyclic since order of G/H is 2 and every group of prime order is cyclic. 
But G is not cyclic. 
I am doing graduation course. 
Please give me an example to disprove or proof for above problem.

REPLY [0 votes]: As a counter-example, you can take $G = C_2\times C_2$.
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