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Let $N$ be finite abelian $p$-group (is not cyclic). Is there any extension (not central extension) of $N$ by $A_{5}$?

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An idea: since you don't want a central extension, direct products are out. The next natural candidate is a non-trivial semidirect product. Since $\,Aut(A_5)\cong S_5\,$ we have quite some possibilities here for some non-cyclic abelian p-groups. My first thought goes to the Klein viergrup. –  DonAntonio Oct 22 '12 at 4:47
    
You also can consider $N$ is non-abelian. –  Ros Oct 22 '12 at 5:44
    
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When you say an "extension of $N$ by $A_5$", do you mean: (i) a group with $G$ with a normal subgroup $K$ with $K \cong N$ and $G/K \cong A_5$; or (ii) a group with a normal subgroup $K$ with $K \cong A_5$ and $G/K \cong N$. Unfortunately, both meanings arise in the literature, with roughly equal frequency! I am guessing you mean (i), since otherwise there would be possibility of it being a central extension. Also is $N$ intended to be a specific abelian $p$-group or an arbitrary abelian $p$-group. In the first case, the answer is that it depends on $N$. –  Derek Holt Oct 22 '12 at 8:25
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2 Answers

The easiest way to construct such examples is to take a nonzero irreducible module $M$ for $A_5$ over the field of order $p$, and let $G$ be the semidirect product of $M$ by $A_5$.

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Here's a big database of them.

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