---
tag: 7YE4TAG5GE
kind: concept
name: Extraspecial group
msc: [20D15]
specializes: [06QYRTX9KQ]
sources:
  - url: https://en.wikipedia.org/wiki/Extraspecial_group
same_as:
  wikidata: Q5422174
  wikipedia: Extraspecial group
---

## Summary

An extraspecial group is a finite $p$-group whose centre has order $p$ and whose quotient by the centre is a nontrivial elementary abelian $p$-group.
They are non-abelian $p$-groups that are as close to abelian as possible, and their order is always an odd power of $p$.
They occur often inside larger groups, for example in centralisers of involutions in sporadic simple groups.

## Explanations

### L1 human

An extraspecial group is almost abelian.
All commutators lie in a tiny central subgroup of order $p$, and the group modulo that subgroup is abelian with every element of order $p$.
The quaternion group $Q_8$ and the dihedral group of order $8$ are the two smallest examples, both with $p = 2$.
For an odd prime $p$, the group of $3 \times 3$ upper unitriangular matrices over the field $\F_p$ is an example of order $p^3$.