Generalized dihedral group

Definition

Suppose $H$ is an abelian group. The generalized dihedral group corresponding to $H$ is the external semidirect product of $H$ with the cyclic group of order two, with the non-identity element acting as the inverse map on $H$ .

Viewing this external semidirect product as an internal semidirect product, $H$ is an abelian normal subgroup of index two.

A presentation for $G$ is:

$G : = ⟨ H, x ∣ x h x^{- 1} = h^{- 1} \forall h \in H ⟩$ .

Note that the dihedral groups are special cases of generalized dihedral groups where the abelian group in question is a cyclic group.

Function	Value	Explanation
order	Twice the order of $H$
exponent	Least common multiple of $2$ and the exponent of $H$
Derived length	$1$ if $H$ is an elementary abelian $2$ -group, $2$ otherwise.
Nilpotency class	Frattini length of $H$ if $H$ is a $2$ -group, not defined otherwise.
Max-length	One more than the max-length of $H$ .
Composition length	One more than the composition length of $H$ .
Chief length	One more than the chief length of $H$ .
Minimum size of generating set	One more than the minimum size of generating set of $H$ .

Property	Satisfied	Explanation
Abelian group	True only if $H$ is an elementary abelian $2$ -group
Nilpotent group	True only if $H$ is a $2$ -group
Solvable group	Yes
Metabelian group	Yes

There are two kinds of subgroups of the generalized dihedral group $G$ with the abelian subgroup $H$ :

Subgroups of $H$ : All of these are normal subgroups of $G$ . The number of such subgroups equals the number of subgroups of $H$ .
Subgroups of $G$ containing an element outside $H$ : These are classified by the following two pieces of information: the choice of a subgroup of $H$ , and the choice of a coset of that subgroup in $G ∖ H$ . The number of such subgroups equals the sum of subgroup indices of $H$ .