Find unique groups of form $$ C_{11}\rtimes C_5$$ (semi-direct product) with homomorphism $$h:C_5\rightarrow Aut(C_{11})$$
I've found the possible homomorphisms i.e $$h=Id,x^3,x^4,x^5,x^9 $$ So performing (1,y)*(x,1) calculation we'll get $$C_{11}XC_5$$ and groups with relations $$YX=X^3Y,YX=X^4Y,YX=X^5Y,YX=X^9Y$$ Are these five groups distinct? Have I done this correctly?
Up to isomorphism, there are only two such groups. One is the abelian one. So you have to show that the remaining four are isomorphic.
To see this, consider the group defined by the first relation: $$ \langle X, Y : X^{11} = Y^5 = 1, Y X Y^{-1} = X^{3} \rangle. $$ Take $Z = Y^{2}$. Then $$ Z X Z^{-1} = Y^{2} X Y^{-2} = Y X^{3} Y^{-1} = (Y X Y^{-1})^{3}= X^{9}. $$ So the group above is the same as $$ \langle X, Z : X^{11} = Z^5 = 1, Z X Z^{-1} = X^{9} \rangle, $$ that is, the group defined by the fourth relation.
Do the same with $Z = X^{3}$ (you will find the group defined by the third relation) and $Z = X^{4}$ (you will find the group defined by the second relation).