Equivalence of presentations of dicyclic group

Statement
Consider the group with presentation:

$$G := \langle a,b,c \mid a^n = b^2 = c^2 = abc \rangle$$

Then:


 * The element $$z = a^n = b^2 = c^2 = abc$$ is in the center of $$G$$.
 * $$z^2$$ is the identity element of $$G$$.
 * If we set $$a = a, x = b^{-1}$$, and denote by $$e$$ the identity element, $$G$$ satisfies the relations $$a^{2n} = e, x^2 = a^n, xax^{-1} = a^{-1}$$. Further, the presentation:

$$\langle a,x \mid a^{2n} = e, x^2 = a^n, xax^{-1} = a^{-1} \rangle$$

also defines $$G$$.

Facts used

 * 1) uses::Group acts as automorphisms by conjugation

Proof
We use the notation as in the statement above.