Dicyclic 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:

$$\langle a,x \mid a^{2n}=e, x^2 = a^n, xax^{-1} = a^{-1} \rangle$$

Here, $$e$$ is the identity element.


 * It has the following faithful representation as a subgroup of the quaternions: $$a = e^{i\pi/n}, x = j$$.
 * It is the binary von Dyck group with parameters $$(n,2,2)$$, i.e., it has the presentation:

$$\langle a,b,c \mid a^n = b^2 = c^2 = abc \rangle$$.

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.

Arithmetic functions
Here, the $$n$$ is as in the parametrization. The order of the group is $$4n$$.

For small values
Note that all dicyclic groups are metacyclic and hence supersolvable. A dicyclic group is nilpotent if and only if it is of order $$2^k$$ for some $$k$$. It is abelian only if it has order 4.

Elements
The dicyclic group of order $$4n$$ has $$n+3$$ conjugacy classes. In the discussion below, we use the presentation:

$$\langle a,x \mid a^{2n}=e, x^2 = a^n, x^{-1}ax = a^{-1} \rangle$$

The elements are:


 * 1) The identity element. (1)
 * 2) The unique central non-identity element, which is given by $$a^n = x^2$$. (1)
 * 3) The remaining elements in $$\langle a \rangle$$. There are $$2n - 2$$ of these elements, and they occur in conjugacy classes of size two: each element is conjugate to its inverse. There are thus $$n - 1$$ conjugacy classes of size $$2$$ each.
 * 4) The elements outside $$\langle a \rangle$$ come in two conjugacy classes: the conjugacy class of $$x$$, which contains all elements of the form $$a^{2k}x$$, and the conjugacy class of $$ax$$. These two conjugacy classes are related by an outer automorphism and each has $$n$$ elements.

Internal links

 * Subgroup structure of dicyclic groups
 * Linear representation theory of dicyclic groups