Hasse norm principle for $M_{11}$ and $J_1$ extensions

We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K\mathrm{/}k$ when the Galois groups $\mathrm{G}\mathrm{a}\mathrm{l}(L\mathrm{/}k)$ of the Galois closure $L\mathrm{/}k$ of $K\mathrm{/}k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,\mathrm{P}\mathrm{i}\mathrm{c}\bar{X})=0$ or $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ for norm one tori $T=R_{K\mathrm{/}k}^{(1)}({\mathbb{G}}_m)$ with a smooth $k$-compactification $X$ and $\bar{X}=X{\times }_k\bar{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups.