What are all nilpotent subgroups of the binary icosahedral group $SL(2,5)$, up to isomorphism? I need this for some sanity check so I'm currently more interested in what the result is than in how to get to it. But it would be nice to see the way too.
The nilpotent subgroups are:
Cyclic Subgroups: Using a short Python program, I found out that each cyclic subgroup is of order 1,2,3,4,5,6 or 10.
$Q_8$, the Lipschitz units (by Wikipedia).
By http://en.wikipedia.org/wiki/Binary_icosahedral_group, the only other subgroups of $SL(2,5)$ are:
- Binary dihedral groups of order 12 and 20 (not nilpotent by http://groupprops.subwiki.org/wiki/Dicyclic_group).
- The binary tetrahedral group of order 24 (by reading a bit in http://en.wikipedia.org/wiki/Binary_tetrahedral_group I figured out it is not the direct product of its Sylow subgroups, thus it's not nilpotent).
Is this correct?