# Symmetries higher dimensional cube fixing subcubes

Which is the group of rotational isometries of an $n$ dimensional hypercube fixing an $m$ dimensional element (an $m$ dimensional subcube)? I know for example that it is $A_n$ for $m=0$ (symmetries fixing a vertex).

• What does "fixing" mean? Pointwise or as a set? – Alex Degtyarev Nov 17 '14 at 18:26
• I'm curious. Where do your hypercube problems come from? – Joonas Ilmavirta Nov 17 '14 at 18:27
• As a set. The problems come from a project i'm working on, but i'm quite confused. – Giovanni_M Nov 17 '14 at 18:32
• Have you tried realizing the group of symmetries as the group of signed permutation matrices with determinant 1? In that language these subgroups should be easy to describe in terms of a block decomposition. – Nate Nov 17 '14 at 19:08
• In these terms, i guess fixing an $m$ subcube means asking that the isometry permutes the $n−m$ faces of the hypercube whose intersection is that subcube. In terms of permutation matrix, it means that there should be two blocks: one of dimension $n−m$, whose entries must be $1$, and one of dimension $m$. That means that the requested group is a subgroup of index $2$ (for the determinant $=1$ condition) of $S_{n-m} \times (\mathbb{Z}_2 \wr S_m)$. Am i missing something? And, what exactly that group is? – Giovanni_M Nov 17 '14 at 20:23