# Presentation for Binary Icosahedral Group "Using" Presentation for $A_5$

This is probably a stupid question, but, per this post Group presentation of $A_5$ with two generators, a presentation for $$A_5$$ is given by $$A_5 \cong \langle x,y \mid x^5=y^2=(xy)^3=1 \rangle$$. In this post Group presentation for semidirect products, it is shown how to make a presentation by generators and relators for semi-direct product if one knows a presentation for the quotient group, the kernel group, and the outer action of the quotient group on the kernel group.

It is known that the binary icosahedral group $$B$$ is a group extension of $$A_5$$ by $$\mathbb{Z}_2$$, $$1 \to \mathbb{Z}_2 \to B \to A_5 \to 1$$. How could one make a presentation by generators and relators for $$B$$ using the presentations $$\mathbb{Z}_2 \cong \langle z \mid z^2=1 \rangle$$ and $$A_5 \cong \langle x,y \mid x^5=y^2=(xy)^3=1 \rangle$$?

In general, if $$1 \to K \to E \to Q \to 1$$, how could one make a presentation by generators and relators for $$E$$ using the presentations for $$K$$ and $$Q$$ if one knew the outer action of $$Q$$ on $$K$$ and one had a set-theoretic section $$\sigma: Q \to E$$ which gave rise to a factor set $$[,]: Q \times Q \to Z(K)$$ which induced the correct element of $$H^2[Q; Z(K)]$$ for the group extension, per the following paper?

https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Ho.pdf

If the extension is non-split, then relations $$r=1$$ of $$Q$$ become $$r=w_K(r)$$ for some word $$w_K(r)$$ in the generators of $$K$$, which have to be determined. You could in principal calculate $$w_K(r)$$ from the cocycle of the extension.
In your example, we can take $$C_2 = \langle z \mid z^2 = 1 \rangle$$. In fact $$w_K(r)=z$$ for each relation $$r=1$$ of $$A_5$$ works (but there are other possible choices). Then you get $$\langle x,y,z \mid x^5=y^2=(xy)^3=z, z^2=1, xz=zx, yz=zy \rangle$$ as a presentation of the binary icosahedral group. In fact the relations $$xz=zx$$ and $$yz=zy$$ are redundant, because $$z$$ is a power of both $$x$$ and of $$y$$, so you can leave those out if you want to.