A **dicyclic group** of order $4n$, denoted $\mathrm{Dic}_n$ or $Q_{4n}$, is a non-split extension of a cyclic group of order $2n$ by a cyclic group of order $2$. That is, $\mathrm{Dic}_n$ has a subgroup of index $2$ which is cyclic, but it is not a semidirect product of that subgroup and a subgroup of order $2$.

The dicyclic group of order $4n$ can be given by generators and relations: $ \langle a,b \mid a^{2n}, a^n b^{-2}, bab^{-1}a\rangle$.

The group $\textrm{Dic}_2$ is isomorphic to the quaternion group of order $8$.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by John Jones on 2019-05-23 19:58:56

**Referred to by:**

**History:**(expand/hide all)

**Differences**(show/hide)