Subgroup ($H$) information
| Description: | $D_8$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(99\)\(\medspace = 3^{2} \cdot 11 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b^{99}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $D_{792}$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{99}.C_{60}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $D_8$ | ||
| Normal closure: | $D_{792}$ | ||
| Core: | $C_8$ | ||
| Minimal over-subgroups: | $D_{88}$ | $D_{24}$ | |
| Maximal under-subgroups: | $C_8$ | $D_4$ | $D_4$ |
Other information
| Number of subgroups in this conjugacy class | $99$ |
| Möbius function | $0$ |
| Projective image | $D_{396}$ |