Number of non isomorphic groups in Cext(G,C_p)

Question

I want to modify my question mathoverflow.net/questions/50922. Let $C_{p^e}$ be a cyclic group of order $p^e$, $p$ prime. Denote by $\text{Cext}(G,C_p)$ the group of all central extensions of $C_p$ by $G$, that is extensions with central subgroup $C_p$ and the quotient $G$. Pick $G=C_{p^e}^n$ for some natural $e,n$. What is the number ,say $I(n,e,p)$, of non isomorphic groups in $\text{Cext}(G,C_p)$?

Answer 1

The answer is very similar to that of http://www.mathoverflow.net/questions/167446/ which was the case $e=1$. In fact it is a little more straightforward when $e>1$, because $p=2$ is no different from any other $p$. (This is because the $(xy)^{p^2} = x^{p^2}y^{p^2}$ for any $x,y$ in a group of this type, whereas $(xy)^p =x^py^p$ holds only when $p$ is odd.)

So assume that $e>1$.

When $n=2$, there are two isomorphism classes of nonabelian groups, one with exponent $p^e$ and the other with exponent $p^{e+1}$. They have presentations

$\langle x,y,z \mid [x,z]=[y,z]=1, [x,y]=z, x^{p^e}=1, y^{p^e}=1, z^p=1 \rangle$ and $\langle x,y,z \mid [x,z]=[y,z]=1, [x,y]=z, x^{p^e}=z, y^{p^e}=1, z^p=1 \rangle$.

By taking central products of these, amalagamating the subgroup $\langle z \rangle$ of order $p$ you get two classes of groups $A_{k,e}$, $B_{k,e}$ of order $p^{2ke+1}$ for each $k \ge 1$, again one with exponent $p^e$ and the other with exponent $p^{e+1}$.

Taking a central product of either of these with a cyclic group of order $p^{e+1}$ (again amalgamating the group of order $p$), results in isomorphic groups $C_{k,e}$.

A general nonabelian group of the type you describe is isomorphic to a direct product of one of $A_{k,e}$, $B_{k,e}$ or $C_{k,e}$ with a direct product of copies of $C_p^e$.