Plus and minus type of an extraspecial $p$-group

Let $p$ be a prime and $n\ge 1$. By For odd $p$ and each $n\ge 1$ there are exactly two extraspecial groups of order $p^{1+2n}$, distinguished by their exponent for odd $p$ and by For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$ for $p=2$, there are exactly two extraspecial groups of order $p^{1+2n}$ up to isomorphism. They are named as follows.

For odd $p$, write $p_{+}^{1+2n}$ for the one of exponent $p$ and $p_{-}^{1+2n}$ for the one of exponent $p^2$ (The exponent of a finite group). At $n=1$ these are the Heisenberg group and the modular group (The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$, The modular group of order $p^3$ is extraspecial, of exponent $p^2$ when $p$ is odd).

For $p=2$, write $2_{+}^{1+2n}$ for the one with $2^{2n}+2^n$ solutions of $x^2=1$ and $2_{-}^{1+2n}$ for the one with $2^{2n}-2^n$ solutions. At $n=1$ these are $\mathrm{Dih}(C_4)$ and $Q_8$ ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively).

The two names are well defined because in each case the two classification theorems supply exactly two isomorphism classes and an invariant that separates them, so the label is a property of the isomorphism class and not of a presentation.

The two source conventions differ in scope and are both recorded here. van Beek's Definition 2.38 writes $p_{\pm }^{1+2n}$ for every prime, the sign being read off an iterated central product, which is the convention taken above. Craven's Definition 3.3 introduces $p_{+}^{1+2}$ only for the odd exponent-$p$ group of order $p^3$ and names the others by their constructions. Where the two overlap they agree, and the convention in force here is van Beek's.

At $p=2$ the signs multiply under central products, as the counting formula shows. At odd $p$ the notation records exponent instead: a central product is of plus type precisely when every order-$p^3$ factor is Heisenberg. If a modular factor occurs, the absorption lemma reduces all modular factors to one, so the product is of minus type. Thus the two uses of the signs agree on the basic plus factors but obey different product rules.