Subgroup ($H$) information
| Description: | $Q_8\times C_{31}$ |
| Order: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(124\)\(\medspace = 2^{2} \cdot 31 \) |
| Generators: |
$ab, b^{24}, b^{16}, c$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{31}:Q_{64}$ |
| Order: | \(1984\)\(\medspace = 2^{6} \cdot 31 \) |
| Exponent: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{248}.C_{60}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $S_4\times C_{30}$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_4\times C_{30}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_{62}$ | ||
| Normalizer: | $Q_{16}\times C_{31}$ | ||
| Normal closure: | $C_{31}\times Q_{32}$ | ||
| Core: | $C_{124}$ | ||
| Minimal over-subgroups: | $Q_{16}\times C_{31}$ | ||
| Maximal under-subgroups: | $C_{124}$ | $C_{124}$ | $Q_8$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_{31}:D_{16}$ |