What are the properties of a dicyclic group?

[Greg Bernhardt]

Definition/Summary

The dicyclic group or generalized quaternion group Dic(n) is a nonabelian group with order 4n that is related to the cyclic group Z(2n).

It is closely related to the dihedral group.

Equations

It has two generators, a and b, which satisfy
$a^{2n} = e ,\ b^2 = a^n ,\ bab^{-1} = a^{-1}$

Its elements are
$Dic_n = \{a^k, ba^k : 0 \leq k < 2n \}$

Its "reflection-like" elements all have order 4, unlike the similar elements of the dihedral group with order 2.
$(ba^k)^4 = e$

Extended explanation

This group may be realized as the matrices
$a^k = \begin{pmatrix} \cos\theta_k & - \sin\theta_k \\ \sin\theta_k & \cos\theta_k \end{pmatrix}$
$ba^k = \begin{pmatrix} i \cos\theta_k & - i \sin\theta_k \\ - i \sin\theta_k & - i \cos\theta_k \end{pmatrix}$
where
$\theta_k = \frac{\pi k}{n}$

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[fresh_42]

Dicyclic groups must not be confused with dihedral groups. Here is an interesting graphic how they are related: https://en.wikipedia.org/wiki/Dicyclic_group#Properties

See also: https://www.physicsforums.com/threads/what-is-a-dihedral-group.762866/