I'm working on a problem where I have to determine the number of 2- and 3-Sylow subgroups in the group of symmetries of the cube in R^3, which has order 48.
What I know thus far is that we can have either 1 or 3 2-Sylow subgroups and 1, 4, or 16 3-Sylow subgroups. Moreover, I also know that a group of order 48 is not simple, so it has proper normal subgroups (however, that doesn't mean any of the Sylow subgroups have to be normal).
How do I narrow down the cases and get to the exact number of 2- and 3-Sylow subgroups? Thanks!