Subgroup ($H$) information
| Description: | $Q_{32}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(62\)\(\medspace = 2 \cdot 31 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$ab, b^{30}$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, 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.
Quotient group ($Q$) structure
| Description: | $D_{31}$ |
| Order: | \(62\)\(\medspace = 2 \cdot 31 \) |
| Exponent: | \(62\)\(\medspace = 2 \cdot 31 \) |
| Automorphism Group: | $F_{31}$, of order \(930\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Outer Automorphisms: | $C_{15}$, of order \(15\)\(\medspace = 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{248}.C_{60}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1860\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 31 \) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_{62}$ | |
| Normalizer: | $C_{31}:Q_{64}$ | |
| Minimal over-subgroups: | $C_{31}\times Q_{32}$ | $Q_{64}$ |
| Maximal under-subgroups: | $C_{16}$ | $Q_{16}$ |
Other information
| Möbius function | $31$ |
| Projective image | $C_{31}:D_{16}$ |