Subgroup ($H$) information
| Description: | $C_{772}$ |
| Order: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Generators: |
$b^{193}, b^{4}, b^{386}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, and cyclic (hence abelian, elementary ($p = 2,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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: | $C_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Automorphism Group: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{192}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{772}$ | |
| Normalizer: | $C_{772}:C_{16}$ | |
| Complements: | $C_{16}$ $C_{16}$ $C_{16}$ $C_{16}$ | |
| Minimal over-subgroups: | $C_4\times D_{193}$ | |
| Maximal under-subgroups: | $C_{386}$ | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{193}:C_{16}$ |