Element structure of extraspecial groups

This article gives specific information, namely, element structure, about a family of groups, namely: extraspecial group.
View element structure of group families | View other specific information about extraspecial group

This article describes the element structure of extraspecial groups. An extraspecial group of order $p^{1 + 2m}$, with $m \ge 1$ and $p$ a prime number, is a non-abelian group $P$ of that order such that $[P,P] = Z(P) = \Phi(P)$ is a cyclic subgroup of order $p$. We can deduce from this that the quotient group is an elementary abelian group of order $p^{2m}$.

For every prime $p$ and every fixed $m$, there are two isomorphism classes of extraspecial groups of order $p^{1+2m}$, known as the extraspecial group of '+' and '-' type respectively.

Summary

Item	Value
conjugacy class sizes	size 1 ($p$ times), size $p$ ($p^{2m} - 1$ times)
number of conjugacy classes	$p^{2m} + p - 1$ See also number of irreducible representations equals number of conjugacy classes, linear representation theory of extraspecial groups
order statistics	depends on whether it's a + or - type; PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.