The compound of an 11-simplex in an 11-hypercube (as a subset of its vertices) has the Mathieu group M11 as its symmetries: https://mathoverflow.net/q/382940/440, via https://cp4space.hatsya.com/2021/02/08/a-curious-construction-of-the-mathieu-group-m11/

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.