Matrix generators for the Harada-Norton group.
Standard generators for .
Construction of the fourfold cover of the Mathieu group .
A symplectic representation of
We explicitly construct a particular real form of the Lie algebra in terms of symplectic matrices over the octonions, thus justifying the identifications and, at the group level, . Along the way, we provide a geometric description of the minimal representation of in terms of rank 3 objects called cubies.
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