Groups of order 3^2.2^n
This article discusses the groups of order , where varies over nonnegative integers. Note that any such group has a 3-Sylow subgroup of order , and a 2-Sylow subgroup, which is of order . 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||Value||Value||Number of groups of order||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|
There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -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.
|total number of groups|| length 1
|length 2||length 3||length 4||length 5|