Let the rigid points of a matrix group refer to subgroups of it that stabilize a nonzero vector and are maximal among such subgroups.

How many conjugacy classes of rigid points are there under the automorphism group $\mathrm{Co}_0$ of the Leech lattice, and what are they?

I am already aware of the following classes:

• $\mathrm{Co}_2$
• $\mathrm{Co}_3$
• $\mathrm{M}_{24}$
• $2^{11}:\mathrm{M}_{23}$
• $\mathrm{P}\Gamma\mathrm{U}_6(2)$

Let the rigid points of a matrix group refer to subgroups of it that stabilize a nonzero vector and are maximal among such subgroups.

How many conjugacy classes of rigid points are there under the automorphism group $\mathrm{Co}_0$ of the Leech lattice, and what are they?

I am already aware of the following classes:

• $\mathrm{Co}_2$
• $\mathrm{Co}_3$
• $\mathrm{M}_{24}$
• $2^{11}:\mathrm{M}_{23}$
• $\mathrm{P}\Gamma\mathrm{U}_6(2)$
• $3^6:(2\times\mathrm{M}_{11})$

Given an $n$-dimensional point group, its rigid points are defined as the points on a unit ($n-1$)-sphere whose stabilizers are not contained in $O_{n-2}$. For a reflection group, the number of orbits of rigid points is equal to its rank, with each orbit being the vertex set of a convex isotoxal polytope. For other point groups, however, things are more complicated.

How many orbits of rigid points are there under the automorphism group of the Leech lattice?

Let the rigid points of a matrix group refer to subgroups of it that stabilize a nonzero vector and are maximal among such subgroups.

How many conjugacy classes of rigid points are there under the automorphism group $\mathrm{Co}_0$ of the Leech lattice, and what are they?

I am already aware of the following classes:

• $\mathrm{Co}_2$
• $\mathrm{Co}_3$
• $\mathrm{M}_{24}$
• $2^{11}:\mathrm{M}_{23}$
• $\mathrm{P}\Gamma\mathrm{U}_6(2)$

Given an n-dimensional point group, its rigid points are defined as the points on a unit (n-1)-sphere whose stabilizers are not contained in $O_{n-2}$. For a reflection group, the number of orbits of rigid points is equal to its rank, with each orbit being the vertex set of a convex isotoxal polytope. For other point groups, however, things are more complicated.

How many orbits of rigid points are there under the automorphism group of the Leech lattice?

Given an $n$-dimensional point group, its rigid points are defined as the points on a unit ($n-1$)-sphere whose stabilizers are not contained in $O_{n-2}$. For a reflection group, the number of orbits of rigid points is equal to its rank, with each orbit being the vertex set of a convex isotoxal polytope. For other point groups, however, things are more complicated.

How many orbits of rigid points are there under the automorphism group of the Leech lattice?