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The question is to work out the number of semi-direct products for from Q to H, where

$ H = C_{42} , Q = C_{3} $

I did:

$Aut(C_{42}) = C_2 \times C_6 = C_{12}$

3 divides 12 telling us that they're are some semidirect products.

If we let Q = (1, 2, 3). We can see that they're are 3 elements in Q (1, 2 and 3) that divide 12, thus there are 3 semi direct products.

Is this correct reasoning?

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    Yeah, that notion of "divides" is very undefined. In particular, we usually write $C_3=\{0,1,2\}$, not $\{1,2,3\}$ and your definition of "division" seems to depend on the labeling.2012-12-13

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