The Automorphism Groups of the Groups of Order 32p

TL;DR

This paper computes and tabulates automorphism groups of groups of order 32p, providing presentations and insights into their structure, especially for groups with normal Sylow p-subgroups, and extends some results to order 32p^2.

Contribution

It offers the first comprehensive computational analysis of automorphism groups for groups of order 32p and 32p^2, including presentations and structural insights.

Findings

Automorphism groups often have the form Hol(C_p) times an invariant factor.

Explicit presentations for automorphism groups of order 32 groups are provided.

Automorphism groups of order 32p^2 are partially determined based on order 32p results.

Abstract

The results of computer computations determining the automorphism groups of the groups of order 32$p$ for $p \geq 3$ are given in several tables. Presentations for the automorphism groups of the groups of order 32, which in many cases appear as direct product factors in the automorphism groups of order $32p$, are also presented for completeness. Many of the groups of order 32$p$ with a normal sylow $p$-subgroup have automorphism groups of the form: Hol($C_p$)$ \times $Invariant Factor. A suggestion is made as to how one might determine this invariant factor using only information on the automorphism group of the 2-group associated with the group of order 32$p$, and the normal subgroup of the 2-group associated with the extension of the group of order $32p$. Some general comments on the groups of order $32p^2$ and their automorphism groups are made. A few explicit calculations for the groups of order $32p^2$ are reported here. Knowing the automorphism groups for the groups of order $32p$ enables us to explicitly write down the automorphism groups for more than half of the automorphism groups of the groups of order $32p^2$.