i want to calculate the holomorph of a group on SageMath, for example
G=CyclicPermutationGroup(6) $\\$
H=G.holomorph()$\\$
G.list() , H$\\$
and i get this result: ([(), (1,2,3,4,5,6), (1,3,5)(2,4,6), (1,4)(2,5)(3,6), (1,5,3)(2,6,4), (1,6,5,4,3,2)], Permutation Group with generators [(3,4,5,6,7,8), (1,2)(4,8)(5,7)])
So, in this case $G$ is represented as permutations of $\mathbb{S}_6$ but $Hol(G)$ is represented on $\mathbb{S}_8$ (?) or maybe some weird representation that i dont understand... there is a way to get $Hol(G)$ also in $\mathbb{S}_6$? I need to check if certain groups are in $Hol(G)$ , but for example if a want to know:$\\$
G in $Hol(G)$ $\\$
the result is False, obviusly for the different representations Anyone can help me?