Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(6176\)\(\medspace = 2^{5} \cdot 193 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{386}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_{772}:C_{16}$ |
| Order: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Exponent: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{386}:C_{16}$ |
| Order: | \(6176\)\(\medspace = 2^{5} \cdot 193 \) |
| Exponent: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| Automorphism Group: | $C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| Outer Automorphisms: | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{772}:C_{16}$ | |||
| Normalizer: | $C_{772}:C_{16}$ | |||
| Minimal over-subgroups: | $C_{386}$ | $C_4$ | $C_2^2$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{386}:C_{16}$ |