I have been studying the permutation group $S_{n}$ in combination with transitivity and blocks. Which leads to primitive groups.
Now I was wondering if for $n=6$, so $S_{6}$ could have two primitive subgroups of different order?
So I'm looking for $\tau\in{}S_{6}$ working on $B\subseteq{}S_{6}$ such that $\tau{}(B)=B$ or $\tau{}(B)\cap{}B=\emptyset{}$.
A block is trivial if $B=(1;...;6)$ or $|B|=1$.
If H is transitive and has only trivial blocks, it is called $primitive$.
So I'm looking for different subgroups of $S_{6}$ which are transitive, has only trivial blocks and have different order. Can anybody give me some help?
Thanx in advance!
(PS - How can I use accolades?)