# Presentation of extentions of groups

Presentation of a semi-direct products of $N$ by $H$ can be written from presentations of $N$ and $H$. But for other extensions of $N$ by $H$ (cyclic, central etc.), which are not semi direct products, can we write presentation of the extension from presentation of $N$ and $H$?

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In general, you need to know two things: what each relator in H is equal to in N, and what generators of H conjugate generators of N to. For example, suppose I have the extension C_2 -> G -> C_2, with presentations <x | x^2> and <y | y^2>. Suppose I know x^y = x and y^2=x, then my presentation for G is <x,y | x^2, y^2=x, x^y=x>. – Steve D Jan 22 '11 at 7:15
The question is: how is the extension given to you in the first place? – Alex B. Jan 22 '11 at 8:25
This is a duplicate of this (very poorly worded, subsequently closed) question: mathoverflow.net/questions/44631/… , which received a very good answer from Derek Holt. – HJRW Jan 22 '11 at 16:08
As per Holder's program, it is possible construct all groups if we know simple groups. "Semi-direct product" is a nice tool to construct many groups from two known groups; but when we try to find all "p-groups" then semi-direct products are not sufficient. We move towards general extension of groups. There may be many extensions of a group $N$ by $H$, which are isomorphic. But if we have presentations of these extensions, we can determine isomorphism classes easily. Therefore, I would like to know, whether presentation of extension can be written from presentation of $N$ and $H$. – RDK Jan 23 '11 at 4:37
What do you mean by 'it is possible to construct all groups'? How do you want to describe the groups? As presentations? It is clearly possible to list all presentations. On the other hand, the isomorphism problem for groups is unsolvable, so you can't possibly hope to classify groups up to isomorphism. – HJRW Jan 23 '11 at 18:50