Exceptional Algebra and Sporadic Groups at c=12

In earlier works, it was seen that a $\mathbb{Z}\mathrm{/}2$ orbifold of the theory of 24 free two-dimensional chiral fermions admits various sporadic finite simple groups as global symmetry groups when viewed as an $\mathcal{N}=1$, ${\cal N}=2$,or${\cal N}=4$ superconformal field theory. In this note, we show that viewing the same theory as an SCFT with extended $\mathcal{N}=1$ symmetry – where the extension is the same one which arises in string compactification on manifolds of exceptional Spin$(7)$ holonomy – yields theories which have global symmetry given by the sporadic groups $M_{24},C{o}_2$ or $C{o}_3$. The partition functions twined by these symmetries, when decomposed into characters of the Spin(7) algebra, give rise to two-component vector-valued mock modular forms encoding an infinite-dimensional module for the corresponding sporadic groups.