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Concept# Dicyclic group

Summary

In group theory, a dicyclic group (notation Dicn or Q4n, ) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as:
:1 \to C_{2n} \to \mbox{Dic}_n \to C_2 \to 1. ,
More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.
Definition
For each integer n > 1, the dicyclic group Dicn can be defined as the subgroup of the unit quaternions generated by
:\begin{align}
a & = e^\frac{i\pi}{n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n} \
x & = j
\end{align}
More abstractly, one can define the dicyclic group Dicn as the group with the following presentation
:\operatorname{Dic}_n = \left\langle a, x \mid a^{2n} = 1,\ x^2 = a^n,\ x^{-1}ax = a^{-1}\right\rangle.,!
Some things to note which follow

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