I'm considering a group acting on a colored 2x2 block. Where the group element $t$ interchanges the top two blocks. Similarly $b$, $r$, and $l$ interchanges the bottom two blocks, right two blocks, and left two blocks, respectively. Is a group presentation for this group $\langle t,b,r,l: t^2=b^2=r^2=l^2=tbtb=rlrl=trtrtr=rbrbrbb=blblbl=tltltl=e \rangle$? If so is this does this group have a common name, or a simpler presentation? Is this the subgroup of $S_4$ which is generated by all transposes?
I am interested in the cycle index notation of this group.
Notice this group is related to the middle 9 cubits in a Rubiks cube. If anyone could give me the group homomorphism between the two that would be extra credit.