# Maria Elisa Fernandes (University of Aveiro)

## Highest rank of a polytope for $$A_n$$

The existence of a regular polytope with a given automorphism group G can be translated into a group-theoretic condition on a generating set of involutions for G. For G the symmetric group $$S_n$$, the maximum rank of such a polytope is n-1, with equality only for the regular simplex. We prove that the highest rank of a string C-group constructed from an alternating group $$A_n$$ is 0 if n=3, 4, 6, 7, 8; 3 if n=5; 4 if n=9; 5 if n=10; 6 if n=11; and $$\lfloor(n-1)/2\rfloor$$ if $$n >11$$.