Subgroup ($H$) information
| Description: | $C_{58}$ |
| Order: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(58\)\(\medspace = 2 \cdot 29 \) |
| Generators: |
$bc^{116}, c^{8}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,29$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_4.D_{116}$ |
| Order: | \(928\)\(\medspace = 2^{5} \cdot 29 \) |
| Exponent: | \(232\)\(\medspace = 2^{3} \cdot 29 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{58}.C_{14}.C_4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_{28}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(928\)\(\medspace = 2^{5} \cdot 29 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_{116}$ | |
| Normalizer: | $C_{58}:Q_8$ | |
| Normal closure: | $C_2\times C_{58}$ | |
| Core: | $C_{29}$ | |
| Minimal over-subgroups: | $C_2\times C_{58}$ | |
| Maximal under-subgroups: | $C_{29}$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_4.D_{116}$ |