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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 $8$.

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  • Last edited by John Jones on 2019-05-23 19:58:56
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