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In a classification of groups of order $p^3q$, where $p, q$ are distinct primes, when considering non-nilpotent groups with normal Sylow $p$-subgroups they mention that these groups have the form $N \rtimes U$, with $|N|=p^3$ and $U=C_q$. Then there is a case where,

1) $N$ is extraspecial of exponent $p$: Then $Aut(N) \rightarrow Aut(N/ \phi(N)) \cong GL(2,p)$ is surjective. Thus the conjugacy classes of subgroups of order $q$ in $Aut(N)$ correspond to the conjugacy classes of subgroups of order $q$ in $GL(2,p)$.

and,

2) $N$ is extraspecial of exponent $p^2$: Then $Aut(N)$ is solvable and has a normal Sylow $p$-subgroup with $p$-complement of the form $C_{p-1}$. Thus there is atmost one subgroup of order $q$ in $Aut(N)$ and this exists iff $q|(p-1)$.

Can someone please help me to identify how to write the above two groups in notation form. i.e. Can the groups be written as follows using notations:

1) $N \cong C_{p^2} \times C_p$

2) $N \cong C_{p^3}$

What is meant by "extraspecial of exponent". Please help me with this question.

Thanks a lot in advance.

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  • $\begingroup$ You'd need to understand first extraspecial groups though. No? The examples you give for the iso. type of $N$ are abelian groups. $\endgroup$ – the_fox Jan 31 at 4:17
  • $\begingroup$ Yes, thanks @the_fox. But is it possible to mention the groups for the moment. I started to read about extraspecial groups. But please can you mention the groups? $\endgroup$ – Buddhini Angelika Jan 31 at 4:42
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    $\begingroup$ The exponent-$p$ type has shape $(C_p \times C_p) \rtimes C_p$ and the exponent-$p^2$ type has shape $C_{p^2} \rtimes C_p$. I guess this is what you are asking. You should learn about these groups though, as they are quite important! $\endgroup$ – the_fox Jan 31 at 5:12
  • $\begingroup$ Yes, thank you very much @the_fox. $\endgroup$ – Buddhini Angelika Jan 31 at 5:15
  • $\begingroup$ Sorry. the answer was not loaded in my browser earlier and I have read it bit differently. But now its properly loaded and is clear. Thank you very much. :) $\endgroup$ – Buddhini Angelika Jan 31 at 5:17
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An extraspecial $p$-group is a nonabelian group $N$ such that the center $Z(N)$ is cyclic of order $p$ and $N/Z(N)$ is an elementary abelian $p$-group, i.e. it is isomorphic to $C_p^n$ for some $n$. So, an extraspecial group of exponent $p$ is an extraspecial $p$-group whose exponent is $p$, and an extraspecial group of exponent $p^2$ is an extraspecial $p$-group whose exponent is $p^2$. (These are the only possible exponents for an extraspecial $p$-group, since $Z(N)$ and $N/Z(N)$ both have exponent $p$ and so $x^{p^2}=1$ for all $x\in N$.)

The extraspecial groups can be completely classified; you can find more information at the link above. In particular, for $p$ odd, an extraspecial group of order $p^3$ and exponent $p$ is isomorphic to the group of matrices of the form $\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1\end{pmatrix}$ over $\mathbb{F}_p$, and an extraspecial group of order $p^3$ and exponent $p^2$ is a semidirect product $C_{p^2}\rtimes C_p$ where the action of $C_p$ on $C_{p^2}$ is nontrivial.

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  • $\begingroup$ Thank you very much @EricWofsey. Can the group of matrices mentioned be represented by using a notation? $\endgroup$ – Buddhini Angelika Jan 31 at 5:04

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