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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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