Linear representation theory of extraspecial groups

Contents

This article gives specific information, namely, linear representation theory, about a family of groups, namely: extraspecial group.
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This article describes the linear representation theory of extraspecial groups. An extraspecial group of order $p^{1 + 2m}$, with $m \ge 1$ and $p$ a prime number, is a non-abelian group $P$ of that order such that $[P,P] = Z(P) = \Phi(P)$ is a cyclic subgroup of order $p$. We can deduce from this that the quotient group is an elementary abelian group of order $p^{2m}$.

For every prime $p$ and every fixed $m$, there are two isomorphism classes of extraspecial groups of order $p^{1+2m}$, known as the extraspecial group of '+' and '-' type respectively.

Summary

Item Value
degrees of irreducible representations over a splitting field 1 occurs $p^{2m}$ times, $p^m$ occurs $p - 1$ times
number of irreducible representations over a splitting field $p^{2m} + p - 1$
See also number of irreducible representations equals number of conjugacy classes, element structure of extraspecial groups
maximum degree of irreducible representation over a splitting field $p^m$
lcm of degrees of irreducible representations over a splitting field $p^m$
field generated by character values (characteristic zero) $\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive $p^{th}$ root of unity. Note that when $p = 2$, this just becomes $\mathbb{Q}$, indicating that the group is a rational group.
minimal splitting field, i.e., minimal field of realization of irreducible representations (characteristic zero) When $p$ is odd, then $\mathbb{Q}(\zeta_p)$, same as field generated by character values. (See also odd-order p-group implies every irreducible representation has Schur index one).
When $p = 2$, it may be strictly bigger than the field generated by character values.

Particular cases

$m$ $p$ $p^{1 + 2m}$ extraspecial group of '+' type linear representation theory extraspecial group of '-' type linear representation theory degrees of irreducible representations over a splitting field (same for both groups) number of irreducible representations = $p^{2m} + p - 1$
1 2 8 dihedral group:D8 linear representation theory of dihedral group:D8 quaternion group linear representation theory of quaternion group 1,1,1,1,2 (1 occurs 4 times, 2 occurs 1 time) 5
1 3 27 prime-cube order group:U(3,3) linear representation theory of prime-cube order group:U(3,3) M27 linear representation theory of M27 1 occurs 9 times, 3 occurs 2 times 11
2 2 32 inner holomorph of D8 linear representation theory of inner holomorph of D8 central product of D8 and Q8 linear representation theory of central product of D8 and Q8 1 occurs 16 times, 4 occurs 1 time 17
2 3 243 1 occurs 81 times, 9 occurs 2 times 83
3 2 128 1 occurs 64 times, 8 occurs 1 time 65

It is also noteworthy that the extraspecial groups are the only groups of their order that have the given degrees of irreducible representations, i.e., any group with those degrees of irreducible representations must be extraspecial.