The Non-Normal Cyclic Subgroup Graph Associated with Quasidihedral Groups of Order 16

Nur Nabilah Abdul Razak (Mathematical Sciences Studies, Universiti Teknologi MARA Johor Branch)
Nur Idayu Alimon (Mathematical Sciences Studies, Universiti Teknologi MARA Johor Branch)
Adem Kilicman (Mathematical Sciences Studies, Universiti Teknologi MARA)
Nor Haniza Sarmin (Department of Mathematical Sciences, Universiti Teknologi)
Nabilah Fikriah Rahin (Centre for Defence Foundation Studies, Universiti Pertahanan Nasional Malaysia)
I Gede Adhitya Wisnu Wardhana (Department of Mathematics, University of Mataram)

Journal of the Indonesian Mathematical Society
Vol. 32 No. 3 (2026): SEPTEMBER

Let \textsl{G} be a group and \textsl{H} be a non-normal cyclic subgroup of \textsl{G}. The non-normal cyclic subgroup graph, denoted as $\Gamma^{NN}_{H}(G)$, is defined as a directed graph with vertex set elements of \textsl{G} such that for two distinct elements \textsl{x} and \textsl{y} in \textsl{G}, \textsl{x} is the initial vertex and \textsl{y} is the terminal vertex of an edge if their product is in \textsl{H}. In 2020, the idea of the non-normal subgroup graph, which is the extension of the subgroup graph was developed. This research identifies the non-normal cyclic subgroup graphs connected to order 16 quasidihedral groups. Firstly, the Groups, Algorithms and Programming (GAP) software is used to identify the non-normal cyclic subgroups. The definition of $\Gamma^{NN}_{H}(G)$ is then used to calculate the adjacency and direction of the vertices. Lastly, Maple 2016 software will be used to visualize these graphs.