Let A=Z42, B=D8, and G=Aut(A×B). We note A is abelian, so we consider F=AutA(A×B)≤G. We also note Z(A×B)=Z42×⟨r2⟩≅Z52. Since A×1≤Z(A×B) char A×B we observe [G : F]=[AutA×B(Z(A×B)) : AutA×BA(Z(A×B))].
Let Φ∈G. Then we note Φ(r2)=Φ(r)2 so since r2 and also Φ(r2) are in Z(A×B) we have Φ(r2)=r2 (r2 is the only nonidentity square in Z(A×B)). Conversely, let σ∈Aut(Z(A×B)) be such that σ(r2)=r2. We observe the relations of A×B generated in ⟨a,b,c,d,s,r⟩:a2=b2=c2=d2=s2=r4=1aba−1b−1=aca−1c−1=ada−1d−1=asa−1s−1=ara−1r−1=...=1(rs)2=1where the second line lists all the commutation relations (all commute but r, s). Some extending map σ′ defined by its action on a,...,r,s is an automorphism iff σ′(a), ... , σ′(r),σ′(s) generate A×B and σ′ is one on these relations generators in the free group. If we let σ′ act on a, b, c, d as σ does and set σ′(s)=s and σ′(r)=r we observe σ′(r2)=σ(r2)=r2 and it follows that σ′ extends σ, and is an automorphism since the orders line is fulfilled (Z(A×B) has exponent 2, so σ′(a)2=σ(a)2=1 and similar for b,c,d and clearly σ′(s)2=σ′(r)4=1), the commutation line is fulfilled (all of these relations involve a, b, c, or d, and σ′ preserves their centricity), the last relation is fulfilled (σ′ fixes D8), and the generation is established as a,...,d is in the image of σ on Z(A×B), hence σ′ on A×B, plus s=σ′(s) and r=σ′(r). Thus AutA×B(Z(A×B)) is isomorphic to the group of isomorphisms fixing the last coordinate of Z52, whose order we compute (25−21)(25−22)(25−23)(25−24)=322,560.
Utilizing the previous reasoning we note that σ∈AutA(Z(A×B)) extends iff σ(r2)=r2, so |AutA×BA(Z(A×B))| is the number of automorphisms of Z52 fixing the last coordinate and restricting to an automorphism of Z42×1, i.e. the number of automorphims of Z42 which is computed to be (24−20)(24−21)(24−22)(24−23)=20,160. Thus we calculate [AutA×B(Z(A×B)) : AutA×BA(Z(A×B))]=16 distinct cosets of AutA(A×B) in Aut(A×B).
We have seen |AutA(A×B)|=|Aut(A)|·|Hom(B,A)|·|Aut(B)|. We have already computed |Aut(Z42)|=20,160. As well, utilizing the relations for D8 yields |Aut(D8)|=8. We see Hom(B,A) is the set of all maps defined freely on r and s in the free group factoring through the relations of D8, i.e. mapping one on the relations of D8, and since Z42 has exponent 2 we see every such mapping is a homomorphism, so |Hom(B,A)|=162=256. |Aut(A×B)|=|G|=[G : F]·|F|=[G : F]·|Aut(A)|·|Hom(B,A)|·|Aut(B)|=16·20,160·256·8=660,602,880
Alternatively, we calculate the number of automorphisms of the characteristic center that extend |AutA×B(Z42×⟨r2⟩)|=322,560 (see above) which represent the cosets of the subgroup of automorphisms that fix the center. We see that sending r to either r or r3 with any coordinates for a, b, c, d together with any mapping of s to s, rs, r2s, or r3s and any coordinates of a, b, c, d are all automorphisms that fix the center, so 2,048·322,560=660,602,880.
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