Linear representation theory of extraspecial groups

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.

Particular cases
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.