Subgroup ($H$) information
| Description: | $C_{496}$ |
| Order: | \(496\)\(\medspace = 2^{4} \cdot 31 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(496\)\(\medspace = 2^{4} \cdot 31 \) |
| Generators: |
$b^{24}, b^{16}, b^{28}, c, b^{30}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,31$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{248}.C_{60}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{60}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2\times C_{60}$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(992\)\(\medspace = 2^{5} \cdot 31 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{496}$ | ||
| Normalizer: | $C_{31}:Q_{64}$ | ||
| Minimal over-subgroups: | $C_{31}\times Q_{32}$ | $C_{31}:Q_{32}$ | $C_{31}:C_{32}$ |
| Maximal under-subgroups: | $C_{248}$ | $C_{16}$ |
Other information
| Möbius function | $2$ |
| Projective image | $C_{31}:D_{16}$ |