# Internal semidirect product and roots of unity

## Homework Statement

Let ##G=G_{12}##, ##H_1=G_3##, ##H_2=G_2##. Decide if there are groups ##K_1##, ##K_2## such that ##G## can be expressed as the internal semidirect product of ##H_i## and ##K_i##.

## The Attempt at a Solution

Suppose I can express ##G_{12}## as an internal semidirect product between ##G_3## and ##K##, with ##K## subgroups. Then it must be ##G_3 \cap K={1}## and ##G_12=G_3.K##. I know that for ##n,m##, ##G_{n} \cap G_{m}={1}## if and only if ##(n:m)=1##. Taking this condition into account and the fact that ##K## has to be a subgroup of ##G_{12}##, the candidates for ##K## would be ##G_i## with ##i \in \{2,4\}##. I am not so sure how to realize if one of these subgroups could work. The case ##H_2=G_2## is analogous, I would appreciate suggestions.

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