Geodesic
The simplest overgroups of regular permutation representations of nonabelian $2$-groups with a cyclic subgroup of index $2$
Matematičeskie voprosy kriptografii, Tome 13 (2022) no. 3, pp. 107-130 
B. A. Pogorelov; M. A. Pudovkina. The simplest overgroups of regular permutation representations of nonabelian $2$-groups with a cyclic subgroup of index $2$. Matematičeskie voprosy kriptografii, Tome 13 (2022) no. 3, pp. 107-130. http://geodesic.mathdoc.fr/item/MVK_2022_13_3_a6/
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     author = {B. A. Pogorelov and M. A. Pudovkina},
     title = {The simplest overgroups of regular permutation representations of nonabelian $2$-groups with a cyclic subgroup of index~$2$},
     journal = {Matemati\v{c}eskie voprosy kriptografii},
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     year = {2022},
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Voir la notice de l'article provenant de la source Math-Net.Ru

For any nonabelian $2$-group $H_m$ with a subgroup of index $2$ (namely the dihedral group $D_{2^m}$, the generalized quaternion group $Q_{2^m}$, the modular maximal-cyclic group $M_{2^m}$, the quasidihedral group $SD_{2^m}$) we consider its simplest overgroups. In this way we describe properties of the group generated by the right and the left regular permutation representations of any $H_m$ including its structure, order, center, rang and estimate of the minimal degree. We characterise its automorphism group and all isomorphic embeddings of $H_m$ (of order $2^m$) into the affine group of the residue ring $\mathbb{Z}_{2^{m - 1}}$ if such embeddings exist.

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