Extraspecial group

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties

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