The Automorphism Groups of the Groups of Order 32p

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 $32p^2$.