Automorphisms of non-abelian groups of order 27 What are the automorphism groups of non-abelian groups of order 27? (there are two non-abelian groups of order 27). 
 A: The non-abelian group of order $p^3$ with no elements of order $p^2$ is the Sylow $p$-subgroup of $\operatorname{GL}(3,p)$.  Its automorphism group can also be viewed as a group of $3\times3$ matrices, the affine general linear group, 
$$\operatorname{AGL}(2,p) = \left\{ \begin{pmatrix}a & b& e\\ c& d& f\\ 0 & 0 & 1\end{pmatrix} : a,b,c,d,e,f \in \mathbb{Z}/p\mathbb{Z},\; ad-bc ≠ 0 \right\}, $$ 
which is the semi-direct product of $\operatorname{GL}(2,p)$ on its natural module.
This description is reasonably famous, especially when considering non-abelian groups of order $p^{2n+1}$ with no elements of order $p^2$ whose center and derived subgroup have order $p$.  Instead of $\operatorname{GL}(2,p)$ you get a variation on $\operatorname{Sp}(2n,p)$, that simplifies to $\operatorname{GL}(2,p)$ when $n=1$.
The non-abelian group of order $p^3$ with an element of order $p^2$ and $p ≥ 3$ has as its automorphism group a semi-direct product of $\operatorname{AGL}(1,p)$ with the dual of its natural module, so you get all $3×3$ matrices
$$\left\{ \begin{pmatrix}a & b& 0\\ 0& 1& 0\\ c & d & 1\end{pmatrix} : a,b,c,d \in \mathbb{Z}/p\mathbb{Z},\; a ≠ 0 \right\}. $$ 
In both cases the "module part" of the semi-direct product is the group of inner automorphisms and the quotient ( $\operatorname{GL}(2,p)$ and $\operatorname{AGL}(1,p)$ ) are the outer automorphism groups.
You can read about some of this in section A.20 of Doerk–Hawkes, or Winter (1972).


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*Winter, David L.
“The automorphism group of an extraspecial p-group.”
Rocky Mountain J. Math. 2 (1972), no. 2, 159–168.
MR297859

*Doerk, Klaus; Hawkes, Trevor.
Finite soluble groups.
de Gruyter Expositions in Mathematics, 4. Walter de Gruyter & Co., Berlin, 1992. xiv+891 pp. ISBN: 3-11-012892-6
MR1169099
