# Element structure of extraspecial groups

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: extraspecial group.
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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]