The automorphism groups of the groups of orders $16p$ and $16p^2$

TL;DR

This paper computes the automorphism groups of groups with orders 16p and 16p^2, addressing specific cases and providing explicit presentations where possible, advancing understanding of their structure.

Contribution

It provides new calculations and explicit presentations of automorphism groups for certain groups of orders 16p and 16p^2, especially handling complex cases involving direct products and specific primes.

Findings

Automorphism groups computed for most cases of order 16p and 16p^2.

Explicit presentations provided for automorphism groups in many instances.

Complex cases involving (C_p × C_p) @ G[16] remain partially unresolved.

Abstract

Results of the computation of the automorphism groups for the groups of orders $16p$ and $16p^{2}$ are given. In some cases it has not been possible to give as complete a set of results as was done previously for the case of groups of order $8p^2$. Problems arise for those groups of the form ($C_{p} \times C_{p}$) @ $\G$[16] that occur in the orders $p\equiv 1$ mod(8) and $p\equiv 7$ mod(8), where $G$[16] means any group of order 16. The groups $G$[16] in question are $C_{16}$, $D_{8}$, $QD_{8}$, and $Q_{4}$. For the other cases, explicit presentations are presented for the automorphism groups of the groups of orders 16$p$ and 16$p^2$.