Examples of nilpotent subgroups of $S_n$ which have less than say 10^9 elements?

Question

To make some computational experiments with finite nilpotent group - it would be helpful to know the following:

Question: What are the examples of nilpotent (but not commutative) subgroups in symmetric group $S_n$ which are not so big - say less than $10^9$ elements ?

Preferably given by explicit generators (permutations).

Answer 1

The smallest non-abelian nilpotent group is the dihedral group of order 8 the symmetries of a square. This can be realized as a permutation group by keeping track of what a square symmetry does to vertices. Then a rotation acts on vertices by cycling them: $(1 \; 2\; 3 \; 4)$ and a reflection acts by $(1 \; 2)(3 \; 4)$. These generate a copy of $D_4$ in $S_4$.

If you want to do computations, here are two other sources of examples:

1. Group extensions: a group is nilpotent iff it is an iterated extension by abelian groups. If you're comfortable with semi-direct products, these can lend themselves well to computations.

2. Upper triangular matrices with 1s on the diagonal. In particular, the Heisenberg group. These groups lend themselves well to computations, and give good intuition about what nilpotent groups ``actually" are when you think about how they act on a vector space.

Answer 2

All $p$-groups, i.e., groups whose order is a prime power $p^m$, are nilpotent, so any nonabelian group of order such a power is an example (necessarily $m \geq 3$). Conversely, all finite nilpotent groups are direct products of groups of prime power orders.

So, the nilpotent, nonabelian groups of order $< 2^5$ are:

• Order 8
  • dihedral group, $D_8 \cong \langle (14)(23), (1234) \rangle$
  • quaternion group, $Q_8 \cong \langle (1247)(3685), (1348)(2576) \rangle$
• Order 16: all $9$ nonabelian groups
• Order 24: $C_3 \times D_8$, $C_3 \times Q_8$
• Order 27: $(C_3 \times C_3) \rightthreetimes C_3$, $C_9 \rightthreetimes C_3$

All of these groups can be realized as groups of permutations of $n \leq 12$ objects, except for $Q_{16}$, the generalized quaternion group, which embeds in no symmetric group smaller than $S_{16}$.