Subgroup ($H$) information
| Description: | $C_{31}:Q_{32}$ |
| Order: | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| Index: | \(2\) |
| Exponent: | \(496\)\(\medspace = 2^{4} \cdot 31 \) |
| Generators: |
$b^{24}, b^{16}, c, b^{4}, b^{30}, a$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{248}.C_{60}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{248}.C_{30}.C_2^4$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(119040\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 31 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_{31}:D_{16}$, of order \(992\)\(\medspace = 2^{5} \cdot 31 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_{31}:Q_{64}$ | ||
| Minimal over-subgroups: | $C_{31}:Q_{64}$ | ||
| Maximal under-subgroups: | $C_{496}$ | $C_{31}:Q_{16}$ | $Q_{32}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{31}:D_{16}$ |