The mod-$2$ cohomology of $32\Gamma_3f$

TL;DR

This paper computes the mod-2 cohomology ring of the order 32 group 32Γ_3f, resolving previous discrepancies by explicitly constructing cocycle representatives of the cohomology generators.

Contribution

It provides the first explicit cocycle representatives for the cohomology generators of 32Γ_3f, clarifying prior conflicting computations.

Findings

Explicit cocycle representatives for cohomology generators

Resolution of discrepancies in previous cohomology computations

Complete description of the mod-2 cohomology ring of 32Γ_3f

Abstract

We compute the group cohomology of $32\Gamma_3f$, a certain group of order 32. For this we construct explicit cocycle representatives of the cohomology generators. We thus lay to rest a discrepancy between several published computations of this cohomology ring.