Extraspecial group

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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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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 p^{2r + 1} for some positive integer r. For each r, there are two extraspecial groups (up to isomorphism), denoted the + and - types respectively.

The case p = 2

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.

r Order 2^{2r + 1} 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 D_8 3 quaternion group Q_8 4
2 32 inner holomorph of D8 D_8 * D_8, Q_8 * Q_8 49 central product of D8 and Q8 D_8 * Q_8 50
3 128  ? D_8 * D_8 * D_8, Q_8 * Q_8 * D_8 2326  ? D_8 * D_8 * Q_8, Q_8 * Q_8 * Q_8 2327

Other primes

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Relation with other properties

Weaker properties