The Hamiltonian Circuits in Double Dihedral Group Q12 and the Symmetry Group D8

This paper analyzed all the properties of some non-Abelian finite groups with two generators, and
contain only Abelian and Hamiltonian subgroups. The two exceptional groups D8 and Q12 of orders 16
and 24 respectively, were examined and are completely determined using GAP. The aim was
achieved due to the fact that if a group G contains at least one Hamiltonian subgroup and if all its
subgroups are either Abelian or Hamiltonian, then the group itself is Hamiltonian. We finally generate
some Hamiltonian circuits in the selected groups and the possible number of circuits in each group.