Let $P$ be a special $p$-group $p^{n+m}$. So $P$ will have $p^m$ linear characters. How does one describe (or determine) the other ordinary irreducible characters of $P$ and will they all be faithful? How many ordinary irreducible characters will $P$ have in total. Also, what can we say about the orders of elements in the conjugacy classes of $P$? I am familiar with the case when $P=p^{1+2k}$ is an extra-special $p$-group, then $P$ will have $p^{2k}$ linear characters and $p-1$ faithful ordinary characters of degree $p^k$.
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2$\begingroup$ If $n>1$ then there cannot be any faithful irreducible (complex) representations because the centre is not cyclic. $\endgroup$– Derek HoltCommented Sep 18, 2021 at 22:00
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2$\begingroup$ Also I don't think the character degrees are a fixed function of $p$, $n$ and $m$. In the case $3^{2+4}$, in some examples there are $36$ of degree $3$ and $4$ of degree $9$, and in others there are $8$ of degree $9$. $\endgroup$– Derek HoltCommented Sep 18, 2021 at 22:15
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1$\begingroup$ One interesting case occurs when $P$ admits a group of automorphisms which acts irreducibly on $P/Z(P)$ but trivially on $Z(P)$. Then for every maximal subgroup $M$ of $Z(P)$, $P/M$ is extraspecial (exercise). It follows that every nonlinear ordinary irreducible character of $P$ has degree $p^{m/2}$. $\endgroup$– Richard LyonsCommented Sep 20, 2021 at 17:12
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