# Extraspecial group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter

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## Contents

## Definition

A group of prime power order (or, more generally, any p-group) is termed **extraspecial** if its center, derived subgroup and Frattini subgroup all coincide, and moreover, each of these is a group of prime order (and hence, a cyclic group).

## Classification and particular cases

Any finite extraspecial group has order for some positive integer . For each , there are two extraspecial groups (up to isomorphism), denoted the + and - types respectively.

### The case

All the extraspecial groups are obtained as iterated central products of copies of dihedral group:D8 and quaternion group. The type corresponds to the cases where the quaternion group part occurs an even number of times in the central product and the type corresponds to the case where the quaternion group occurs an odd number of times.

Order | Group of type | Central product expressions | GAP ID (second part) | Group of type | Central product expressions | GAP ID (second part) | |
---|---|---|---|---|---|---|---|

1 | 8 | dihedral group:D8 | 3 | quaternion group | 4 | ||

2 | 32 | inner holomorph of D8 | , | 49 | central product of D8 and Q8 | 50 | |

3 | 128 | ? | , | 2326 | ? | , | 2327 |

### Other primes

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## Relation with other properties

### Weaker properties

- Special group
- Frattini-in-center group
- Group of nilpotency class two
- Camina group:
`For full proof, refer: extraspecial implies Camina`