is the group cyclic, if Normal Subgroup and Quotient group are cyclic.

Question

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.

Answer 1

As a counter-example, you can take $G = C_2\times C_2$.