Groups of order 3^2.2^n

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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 ?