I know that, by D. McKay and C. E. Praeger papers" Vertex-transitive graphs which are not Cayley graphs I", there exist 112 non-Cayley vertex-transitive graph with 24 vertices. 
Is there any such graph of valency 5? Which graph is the answer?



Is there any vertex-transitive non-Cayley graph with 24 vertices and valency 5?
Yes, there is one, see [this paper](https://cs.anu.edu.au/people/Brendan.McKay/papers/Transitive26.pdf). 


You can fetch it from [Gordon Royle's collection](http://staffhome.ecm.uwa.edu.au/~00013890/remote/trans/index.html).


