Subgroup ($H$) information
| Description: | $C_{191}$ |
| Order: | \(191\) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(191\) |
| Generators: |
$b^{128}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $191$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{191}\times Q_{256}$ |
| Order: | \(48896\)\(\medspace = 2^{8} \cdot 191 \) |
| Exponent: | \(24448\)\(\medspace = 2^{7} \cdot 191 \) |
| Nilpotency class: | $7$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Quotient group ($Q$) structure
| Description: | $Q_{256}$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(128\)\(\medspace = 2^{7} \) |
| Automorphism Group: | $C_{64}.C_{32}.C_2^2$, of order \(8192\)\(\medspace = 2^{13} \) |
| Outer Automorphisms: | $C_2\times C_{32}$, of order \(64\)\(\medspace = 2^{6} \) |
| Nilpotency class: | $7$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{190}\times C_{64}.C_{32}.C_2^2$, of order \(1556480\)\(\medspace = 2^{14} \cdot 5 \cdot 19 \) |
| $\operatorname{Aut}(H)$ | $C_{190}$, of order \(190\)\(\medspace = 2 \cdot 5 \cdot 19 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{191}\times Q_{256}$ |
| Normalizer: | $C_{191}\times Q_{256}$ |
| Complements: | $Q_{256}$ |
| Minimal over-subgroups: | $C_{382}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $Q_{256}$ |