Dicyclic group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with metacyclic group

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with dihedral group

Definition

The dicyclic group, also called the binary dihedral group with parameter $n$ is defined in the following equivalent ways:

• It is given by the presentation:

$<a,x|a^{2n}=1,x^2=a^n,x^{-1}ax=a^{-1}>$

• It has the following representation as a subgroup of the quaternions: $a=e^{i\pi /n},x=j$

The dicyclic group with parameter $n$ has order $4n$, and it is an extension of a cyclic group of order $2n$ by a cyclic group of order 2.

Particular cases

• When $n=2$ we get the quaternion group
• When $n$ is a power of $2$ we get a generalized quaternion group