Higher Dimensions
Just as the tetrahedral group generalizes to the rotational symmetry group of the n-simplex (as a subgroup of SO(n)), there is a corresponding higher binary group which is a 2-fold cover, coming from the cover
The rotational symmetry group of the n-simplex can be considered as the alternating group on letters, and the corresponding binary group is a 2-fold covering group. For all higher dimensions except and (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the covering group (maximal cover) and is superperfect, but for dimensional 5 and 6 there is an additional exceptional 3-fold cover, and the binary groups are not superperfect.
Read more about this topic: Binary Tetrahedral Group
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