Q:

>Determine subgroup $\langle (12),(13)(24)\rangle $ of group $ S_{4}$

Determine subgroup $\langle (12),(13)(24)\rangle $ of group $ S_{4}$. 

Using the definition of generator of group, I believe I am supposed to find all permutations that can be written as multiple of $(12),(13)(24)$ and/or their inverses. How do I know I where to stop, because there is a lot of elements to multiply ( for example (12),(13)(24),(1324),(12)(34),(1432) etc. )?

A:

$G = \langle (12),(13)(24)\rangle = \langle (12),(12)(13)(24)\rangle = \langle (12),(3241)\rangle$. Let $s = (12)$ and $r = (3241)$ we have $G = \langle r,s\mid r^4 = s^2 = 1, srs = r^{-1}\rangle\cong D_8$, where $D_8$ denotes the dihedral group of order $8$, and $G = \{1,r,r^2,r^3,s,sr,sr^2,sr^3\}$.

