Stabilizer of a point and orbit of a point I really need help with this topic I have an exam tomorrow and am trying to get this stuff in my head. But the book is not explaining me these two topics properly.
It gives me the definition of a stabilizer at a point where $\mathrm {Stab}_G (i) = \{\phi \in G \mid \phi(i) = i\}$, and where $\mathrm{Orb}_G (i) = \{\phi(i) \mid \phi \in G\}$.
I do not know how to calculate the stabilizer nor the orbit for this. I am also given an example
Let $G = \{ (1), (132)(465)(78), (132)(465), (123)(456), (123)(456)(78), (78)\}$ and then
$\mathrm{Orb}_G (1) = \{1, 3, 2\}$, 
$\mathrm{Orb}_G (2) = \{2, 1, 3\}$,
$\mathrm{Orb}_G (4) = \{4, 6, 5\}$, and 
$\mathrm{Orb}_G (7) = \{7, 8\}$.
also 
$\mathrm{Stab}_G (1) = \{(1), (78)\},\\ \mathrm{Stab}_G (2) = \{(1), (78)\},\\ \mathrm{Stab}_G (3) = \{(1), (78)\},\text {and}\\ \mathrm{Stab}_G (7) = \{(1), (132)(465), (123)(456)\}.$
If someone could PLEASE go step by step in how this example was solved it would be really helpful.
Thank you

In simple terms,
Stabilizer of a point is that permutation in the group which does not change the given point
=> for stab(1) = (1), (78)
Orbit of a point(say 1) are those points that follow given point(1) in the permutations of the group.
=>orbit(1) = 1 for (1); 3 for (132)...; 2 for (123)...
