Equi-isoclinic planes of Euclidean spaces

In a Euclidean space, a p-set of equi-isoclinic planes is a set of p isoclinic planes of which each pair has the same non-zero angle.The Euclidean 4-space E4 contains a unique congruence class of quadruples of equi-isoclinic planes, whereas quintuples of equi-isoclinic planes do not exist in E4.In the following a method is given to derive sets of equi-isoclinic planes in Euclidean spaces. We find again the well-known sets of equi-isoclinic planes of E4. The quadruples of equi-isoclinic planes in E5 are derived. It turns out that E5 contains one congruence class of such quadruples which are not flat quadruples and one congruence class of quintuples of equi-isoclinic planes, whereas sextuples of equi-isoclinic planes do not exist in E5.It appears that the symmetry group of that quintuple is isomorphic to the symmetric group S5.