How to find a base of the permutation group G=⟨x,y⟩≤S4 x=(1,2,3), y=(1,2,4)?
I hear that base for G is a sequence B = [$b_1, ..., b_m$] ⊂ Ω such that the only element of G which stabilizes each $b_i$ is the identity
I am new to Group theory and I read somewhere that the <> notation is for generators so in this case it will be the list of generators formed by the cartesian product as listed below:
<1,1>, <1,2>, <1,4> , <2,1>, <2,2>, <2,4>, <3,1>, <3,2> and <3,4>
And each of these generators will generate a set so should I compute all these generating sets and make sure none of these sets (besides the identity set) stabilizes each $ b_i $ of the base?