Subgroup ($H$) information
| Description: | $Q_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(154\)\(\medspace = 2 \cdot 7 \cdot 11 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ab, b^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $C_{77}:Q_{16}$ |
| Order: | \(1232\)\(\medspace = 2^{4} \cdot 7 \cdot 11 \) |
| Exponent: | \(616\)\(\medspace = 2^{3} \cdot 7 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{77}.C_{30}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $Q_{16}$ | ||
| Normal closure: | $C_{77}:Q_8$ | ||
| Core: | $C_4$ | ||
| Minimal over-subgroups: | $C_{11}:Q_8$ | $C_7:Q_8$ | $Q_{16}$ |
| Maximal under-subgroups: | $C_4$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $77$ |
| Möbius function | $-1$ |
| Projective image | $C_{77}:D_4$ |