Groups of order 2.3^n

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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.

See also groups of order 3^n.

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.