n=12;
r1=Cycles[{{1,2,3,4,5,6}}];
r2=Cycles[{{7,8,9,10,11,12}}];
l1=Cycles[{{2,11},{3,12}}];
l2=Cycles[{{6,9},{5,8}}];
m=Cycles[{{1,2,12,4,5,9},{3,7,8,6,10,11}}];
permIndex[c_Cycles,n_Integer:n]:=Module[{cycleLengths,counts},cycleLengths=Length/@c[[1]];
counts=Table[Count[cycleLengths,k],{k,2,n}];
Prepend[counts,n-Total[cycleLengths]]];

permIndex[PermutationProduct[r1,r1,r1,r1,r1,r1]]

G=PermutationGroup[{r1,r2,l1,l2}];

GroupElementToWord[G,m]

GroupGenerators[G]