A finite group with a non trivial 3 cocycle that is trivial on any cyclic subgroup.

Question

Is there an example of such a 3 cocycle on a finite group? I was looking at the examples of finite abelian groups and odd dihedral groups and there all non trivial 3 cocycles restrict to non trivial cocycles on some cyclic group. Here I am mainly considering $H^3(G,\mathbb{T})$ but I'd be interested in the case of integer cohomology too (Integer cohomology comes out to be trivial by the comments below).

Answer 1

I think an example can be given by letting $G=Q_8$ be the quaternion group. From this paper https://www.tandfonline.com/doi/abs/10.1081/AGB-120005809 we get that $H^4(Q_8,\mathbb{Z})$ is cyclic of order $8$. We know that all cyclic subgroups of $Q_8$ have order $4$ or less, so the groups $H^4(C,\mathbb{Z})$ are all annihilated by multiplication by $4$.

Thus, four times a generator of $H^4(Q_8,\mathbb{Z})$ maps to zero in each $H^4(C,\mathbb{Z})$. This then translates to the torus case via the natural isomorphism $H^3(G,\mathbb{T})\cong H^4(G,\mathbb{Z})$ arising from the short exact sequence $0\rightarrow \mathbb{Z}\rightarrow \mathbb{R}\rightarrow \mathbb{T}\rightarrow 0$.

I should say that I only just looked up the computation of $H^4(Q_8,\mathbb{Z})$, so you should probably double check that paper independently if you want to use this example for anything, in case I made a mistake.