The action of $S_5$ on $6$ points

Question

I wanted to show that the symmetric group on $5$ points ($S_5$ as a subgroup of $S_6$) acting on $6$ points is not a transitive action.

Is this a valid group action?

It is clearly not transitive because there are more points to act on but I am not sure if this is considered as a valid example. thank you

Answer 1

Yes, it is a valid action, provided that $S_5$ fixes an element of the six points, WLOG the sixth one. Verifying that this is an action is routine (and follows from the definition of $S_5$).