You can look at this.
Also, and so it is not dicyclic.
Yes of course , because 11 is prime number.
But maybe I did not say specifically about the question. I meant when you do the semidirect product and crucial calculation.
The actual question is: describe explicitly all homomorphisms of h:C4 --> Aut(C11). This is the short answer ( I cant type all of it)
C4 ={1,y,...,y^3}
C11 = {1,x,.....,x^10}
Thus there is a unique subgroup of order 2 of C11, say t(x) = x^(-1)
So, there are 2 homomorphisms 1 and t
When doing the crucial calculation, we get 2 groups:
< X,Y :X^11 = Y^4 = 1, YX = XY> which is abelian group (1)
< X,Y : X^11 = Y^4 = 1, YXY^(-1) = X^(-1)> (2)
In the lecture, my lecturer said the second group (2) is binary dihedral group.
I hope what I write down is clear to you
Many thanks