# Groups of order 3^2.2^n

This article discusses the groups of order $3^2 \cdot 2^n$, where $n$ varies over nonnegative integers. Note that any such group has a 3-Sylow subgroup of order $3^2 = 9$, and a 2-Sylow subgroup, which is of order $2^n$. Further, because order has only two prime factors implies solvable, any such group is a solvable group and in particular a finite solvable group.

## Number of groups of small orders

Exponent $n$ Value $2^n$ Value $3^2 \cdot 2^n$ Number of groups of order $3^2 \cdot 2^n$ Reason/explanation/list
0 1 9 2 See groups of order 9, classification of groups of prime-square order
1 2 18 5 See groups of order 18
2 4 36 14 See groups of order 36
3 8 72 50 See groups of order 72
4 16 144 197 See groups of order 144
5 32 288 1045 See groups of order 288
6 64 576 8681 See groups of order 576
7 128 1152 157877 See groups of order 1152

## Arithmetic functions

### Derived length

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order. $n$ $3^2 \cdot 2^n$ total number of groups length 1
(abelian groups)
length 2 length 3 length 4 length 5
0 9 2 2
1 18 5 2 3
2 36 14 4 10
3 72 50 6 37 7
4 144 197 10 159 22 6
5 288 1045 14 876 123 32
6 576 8681 22 7625 800 234
7 1152 157877 30 ?