Cardinality of a group Let $G = <a,b,c>$, where $a=(1,2)$, $b=(2,3)$ and $c=(4,5)$ are elements of $S_5$
Basically I had a question to compute the orbit and stabilisers and eventually state the orbit-stabiliser theorem and by that theorem I found the Cardinality of |G| to be 12. 
My question may be trivial but how can I find the cardinality of G without doing that?(as if to confirm I'm correct)

$ab$ and $a$ generate $S_3$, which has 6 elements.
$c$ commutes with $S_3$ and so $G=S_3 \cup cS_3$.
Since $S_3$ and $cS_3$ are disjoint, you get that $G$ has 12 elements.
