The Schur Theorem: if $\left\vert G:Z\left( G\right) \right\vert $ is finite, then $G^{\prime }$ is finite.
My question is: if $1\not=N\trianglelefteq G$ such that $\left\vert G:NZ\left( G\right) \right\vert $ is finite, is there some information on $G^{\prime }$ or
some finiteness conditions involving $G^{\prime }?$