Groups of order 2.3^n

This article discusses the groups of order $2 \cdot 3^{n}$ , where $n$ varies over nonnegative integers. Note that any such group has a normal 3-Sylow subgroup and a complement to that which is a 2-Sylow subgroup isomorphic to cyclic group:Z2. Thus, the group is either a nilpotent group (a direct product of the 3-Sylow subgroup and cyclic group:Z2) or can be described as the internal semidirect product of a group of order $3^{n}$ by a group of order two whose non-identity element acts as a non-identity automorphism.

Number of groups of small orders

Exponent $n$	Value $3^{n}$	Value $2 \cdot 3^{n}$	Number of groups of order $2 \cdot 3^{n}$	Reason/explanation/list
0	1	2	1	cyclic group:Z2, see also equivalence of definitions of group of prime order
1	3	6	2	See groups of order 6. The groups are cyclic group:Z6 and symmetric group:S3.
2	9	18	5	See groups of order 18.
3	27	54	15	See groups of order 54.
4	81	162	55	See groups of order 162.
5	243	486	261	See groups of order 486.
6	729	1458	1798	See groups of order 1458.