How to show that $\mathbb{Z}_2 \wr\mathbb{Z}_3$ is not nilpotent?

Question

There's a Theorem by G.Baumslag (https://www.cambridge.org/core/services/aop-cambridge-core/content/view/92AE6ED711DC037B81E8AC3C30AEE145/S0305004100033934a.pdf/div-class-title-wreath-products-and-span-class-italic-p-span-groups-div.pdf) that says that Wreath Products $A\wr B$are nilpotent if and only if A and B are $p$-groups for the same prime $p$, $B$ finite and $A$ of finite exponent. Clearly, $\mathbb{Z}_2 \wr \mathbb{Z}_3$ doesn't satisfy these conditions, so how can I show that it isn't nilpotent? I tried to find a subgroup that wasn't nilpotent and struggled. Thank you