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What is the automorphism group of direct sum of dihedral group of order $8$, $D_4$?

For example, $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$. How about $\mathrm{Aut}(D_4\times D_4)$, $\mathrm{Aut}(D_4\times D_4\times D_4)$, and $\mathrm{Aut}(D_4\times D_4 \times D_4 \times D_4)$?

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2
  • $\begingroup$ Is $D_4$ the dihedral group of order $8$? $\endgroup$
    – LSpice
    Aug 26 '18 at 23:47
  • $\begingroup$ @LSpice Yes it is. $\endgroup$
    – Sirui Lu
    Aug 26 '18 at 23:47
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The following papers are relevant:

[1] J. N. S. Bidwell, M. J. Curran, and D. J. McCaughan, Automorphisms of direct products of finite groups, Arch. Math. 86, 481 – 489 (2006).

[2] J. N. S. Bidwell, Automorphisms of direct products of finite groups II, Arch. Math. 91, 111–121 (2008).

For your question you want to look at [2]. This paper describes the automorphism group of $G = H^n = H \times \cdots \times H$ where $H$ is an indecomposable non-abelian group. In this case $\operatorname{Aut}(G)$ has a normal subgroup $\mathscr{A}$ isomorphic to the group formed by the matrices $$\left\{ \begin{pmatrix} \alpha_{11} & \cdots & \alpha_{1n} \\ \vdots & \ddots & \vdots \\ \alpha_{n1} & \cdots & \alpha_{nn}\end{pmatrix} : \begin{align}\alpha_{ii} &\in \operatorname{Aut}(H) \text{ for all } 1 \leq i \leq n \\ \alpha_{ij} &\in \operatorname{Hom}(H, Z(H)) \text{ for all i $\neq$ j} \end{align}\right\}.$$

(The group operation is matrix multiplication, with multiplication defined by composition and addition defined by $(\alpha+\beta)(x) = \alpha(x)\beta(x)$.)

Theorem 3.1 of [2] states that $\operatorname{Aut}(G) = \mathscr{A} \rtimes S_n$, where $S_n$ is the symmetric group acting on $G$ by permuting the direct factors. Thus $|\operatorname{Aut}(G)| = |\operatorname{Aut}(H)|^n |\operatorname{Hom}(H, Z(H))|^{n^2-n} n!$

In your case $\operatorname{Aut}(H) \cong D_4$ and $\operatorname{Hom}(H, Z(H)) \cong C_2 \times C_2$, so $|\operatorname{Aut}(G)| = 2^{2n^2+n} n!$.

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2
  • $\begingroup$ Thanks for your answer! Can we get things beyond order, like generators of G? $\endgroup$
    – Sirui Lu
    Aug 27 '18 at 2:39
  • 1
    $\begingroup$ @SiruiLu: Not sure what you mean. If you know $\operatorname{Aut}(H)$ and $\operatorname{Hom}(H, Z(H))$, you have an explicit description of all the automorphisms of $G$, not just the number of them. $\endgroup$ Aug 27 '18 at 2:53
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Mikko Korhonen has already given a good answer. -- But as you asked for explicit generators for the automorphism groups -- you can obtain such by GAP as follows (you see that 4 generators suffice):

gap> D4 := Group((1,2,3,4),(1,3));
Group([ (1,2,3,4), (1,3) ])
gap> A1 := AutomorphismGroup(D4);                  
<group of size 8 with 3 generators>
gap> SmallGeneratingSet(A1);
[ [ (2,4), (1,4)(2,3) ] -> [ (1,2)(3,4), (2,4) ], 
  [ (2,4), (1,4)(2,3) ] -> [ (2,4), (1,2)(3,4) ] ]
gap> A2 := AutomorphismGroup(DirectProduct(D4,D4));
<group of size 2048 with 11 generators>
gap> SmallGeneratingSet(A2);                       
[ [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), 
      (1,2)(3,4)(5,7)(6,8), (1,4)(2,3), (1,3)(2,4)(5,8)(6,7) ], 
  [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), 
      (1,4)(2,3), (1,2)(3,4)(5,7)(6,8), (1,3)(2,4)(5,8)(6,7) ], 
  [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,2)(3,4)(5,7)(6,8), (1,3)(2,4)(6,8), 
      (5,6)(7,8), (1,3)(2,4)(5,8)(6,7), (2,4)(5,7)(6,8) ], 
  [ (1,3)(2,4)(6,8), (2,4)(5,7)(6,8), (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), 
      (1,3)(2,4)(5,8)(6,7) ] -> [ (1,3)(2,4)(5,6)(7,8), (2,4)(5,7)(6,8), 
      (1,2)(3,4), (1,4)(2,3)(5,7)(6,8), (1,3)(2,4)(6,8) ] ]
gap> A3 := AutomorphismGroup(DirectProduct(D4,D4,D4));
<group of size 12582912 with 23 generators>
gap> SmallGeneratingSet(A3);                          
[ [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (2,4)(5,6)(7,8)(10,12), (1,4)(2,3)(5,6,7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,4)(2,3)(6,8)(9,12)(10,11), 
      (1,3)(5,6)(7,8)(9,10)(11,12), (1,4)(2,3)(5,6)(7,8)(9,12,11,10) ], 
  [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (1,3)(5,8)(6,7)(9,10)(11,12), (1,2,3,4)(6,8)(9,11), 
      (2,4)(6,8)(9,10)(11,12), (1,2)(3,4)(5,7)(10,12), (2,4)(5,8)(6,7)(9,11), 
      (2,4)(5,7)(9,10,11,12) ], 
  [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (2,4)(5,8)(6,7)(9,11), (1,4)(2,3)(5,6,7,8)(9,10)(11,12), 
      (2,4)(5,8)(6,7)(9,10)(11,12), (1,4)(2,3)(6,8)(9,10)(11,12), 
      (1,2)(3,4)(5,8)(6,7)(10,12), (1,2,3,4)(5,6)(7,8)(9,12)(10,11) ], 
  [ (1,2)(3,4)(6,8)(9,11), (1,2,3,4)(5,6)(7,8)(9,10)(11,12), 
      (1,2)(3,4)(5,6)(7,8)(10,12), (1,3)(5,6)(7,8)(9,10)(11,12), 
      (1,4)(2,3)(6,8)(9,12)(10,11), (1,2)(3,4)(5,8)(6,7)(9,10,11,12) ] -> 
    [ (1,3)(5,7)(9,10)(11,12), (1,4,3,2)(5,8)(6,7)(10,12), 
      (1,3)(5,6)(7,8)(9,10)(11,12), (1,2)(3,4)(5,6)(7,8)(9,11), 
      (1,3)(5,7)(9,11), (1,3)(5,6)(7,8)(9,12,11,10) ] ]
gap> A4 := AutomorphismGroup(DirectProduct(D4,D4,D4,D4));
<group of size 1649267441664 with 40 generators>
gap> SmallGeneratingSet(A4);                             
[ [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (2,4)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16), 
      (5,6)(7,8)(9,12)(10,11)(13,15), (1,2)(3,4)(6,8)(9,11)(10,12)(13,14)(15,
        16), (1,3)(5,8)(6,7)(9,11)(13,15)(14,16), (2,4)(6,8)(14,16), 
      (1,3)(2,4)(5,6)(7,8)(10,12)(13,16)(14,15), 
      (2,4)(5,8)(6,7)(13,14)(15,16), (1,3)(5,7)(6,8)(9,10)(11,12)(14,16) ], 
  [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (1,3)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16), (1,3)(5,7)(9,10)(11,12), 
      (1,3)(2,4)(5,6)(7,8)(10,12)(14,16), 
      (1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15), 
      (6,8)(10,12)(13,14)(15,16), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,15)(14,
        16), (5,6)(7,8)(9,12)(10,11)(13,14)(15,16), 
      (2,4)(6,8)(9,11)(10,12)(13,16)(14,15) ], 
  [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,14)(15,16), 
      (5,6)(7,8)(9,11)(13,14)(15,16), (2,4)(6,8)(9,10)(11,12)(13,15)(14,16), 
      (1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(14,16), (1,4)(2,3)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(13,15), 
      (1,4)(2,3)(5,8)(6,7)(9,10)(11,12), 
      (1,2)(3,4)(5,7)(6,8)(10,12)(13,16)(14,15) ], 
  [ (2,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15), 
      (5,6)(7,8)(9,11)(13,16)(14,15), (1,2)(3,4)(6,8)(9,10)(11,12)(13,15)(14,
        16), (1,3)(5,8)(6,7)(9,11)(10,12)(13,15), (2,4)(6,8)(10,12), 
      (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(14,16), (2,4)(5,8)(6,7)(9,10)(11,12), 
      (1,3)(5,7)(6,8)(10,12)(13,14)(15,16) ] -> 
    [ (1,3)(2,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16), 
      (1,2)(3,4)(5,8)(6,7)(9,10)(11,12), (1,3)(5,7)(9,11)(10,12)(14,16), 
      (1,3)(2,4)(5,8)(6,7)(9,11)(13,16)(14,15), 
      (1,2)(3,4)(5,7)(9,11)(10,12)(13,16)(14,15), 
      (1,3)(5,6)(7,8)(10,12)(13,15)(14,16), (2,4)(5,8)(6,7)(13,14)(15,16), 
      (1,4)(2,3)(9,12)(10,11)(13,16)(14,15) ] ]
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