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The compound of an 11-simplex in an 11-hypercube (as a subset of its vertices) has the Mathieu group M11 as its symmetries: mathoverflow.net/q/382940/440, via cp4space.hatsya.com/2021/02/08

The cp4space post goes on to describe how to find two dual 11-simplices in the same hypercube from the perfect ternary Golay code, much like the two simplices in a 3-cube that form the stella octangula.

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