Generalized quaternion group

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Definition

A generalized quaternion group is a group of order 2^{k+1} with generators x and a such that the group has the presentation:

<a,x| x^2 = a^{2^{k-1}}, a^{2^k} = 1, xax^{-1} = a^{-1}>

Equivalently, it is the dicyclic group with parameter 2^{k-1}.

For the particular case k=2, we recover the quaternion group.