TITLE: Show that $\langle(1234),(12)\rangle = S_4$
QUESTION [6 upvotes]: I am trying to show that $\langle(1234),(12)\rangle = S_4$. I can multiply and get all $24$ permutations manually but isn't there a more compact solution?

REPLY [3 votes]: I hope you are familiar to $S_4$ presentation:
$$S_4=\langle a,b\mid a^2=b^4=(ab)^3=1\rangle$$ Considering a nice Corollary of Von Dyck's Theorem we can set $a=(1,2),~~~b=(1,2,3,4)$ and see that $(ab)^3=1,a^2=b^4=1$. Also, an small program in GAP's environment could show this fast and easy:
gap> f:=FreeGroup("a","b");
gap> a:=f.1;; b:=f.2;; 
     s:=f/[a^4,b^2,(a*b)^3];;
gap> StructureDescription(s);
                                         "S4"