Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{16}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), stem, a $p$-group, simple, and rational.
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.
Quotient group ($Q$) structure
| Description: | $C_{31}:D_{16}$ |
| Order: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| Exponent: | \(496\)\(\medspace = 2^{4} \cdot 31 \) |
| Automorphism Group: | $C_{124}.C_{60}.C_2^3$ |
| Outer Automorphisms: | $C_2\times C_{60}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{248}.C_{60}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(238080\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \cdot 31 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{31}:Q_{64}$ | |||
| Normalizer: | $C_{31}:Q_{64}$ | |||
| Minimal over-subgroups: | $C_{62}$ | $C_4$ | $C_4$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{31}:D_{16}$ |