$\operatorname{Aut}(K_4 \times \mathbb{Z}_2)$ is isomorphic to group of order 168.
I don't know how to start to find an element of $\operatorname{Aut}(K_4 \times \mathbb{Z}_2)$ such that his order divides $3$ (non-trivial).
I appreciate your help.
2026-03-30 20:44:06.1774903446
Exist an easy way to find a homomorphism from $\mathbb{Z}_3$ to $\operatorname{Aut}(K_4 \times \mathbb{Z}_2)$?
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If $K_4$ is the Klein $4$-group, then $K_4\times Z_2\cong Z_2\times Z_2\times Z_2$. This group has an order $3$ automorphism cycling the three factors, i.e., $(a,b,c)\mapsto(b,c,a)$.