the help on the last post does seem to have helped me a lot atleast upto the point where i can understand the basics.

heres the next one, as usual any hints on how to approach the question would be greatly appreciated. i will attempt the questions with the hints

(1)

describe explicitly all homomorphisms

$\displaystyle \varphi : C_4 \rightarrow Aut(C_5)$

(2)

For each such homomorphism $\displaystyle \varphi$ describe the semidirect product $\displaystyle C_5 \rtimes_{\varphi} C_4$ in terms of generators and relations.

(3) How many distinct isomorphism types of groups of the form $\displaystyle C_5 \rtimes_{\varphi} C_4$ are there?