Subgroup ($H$) information
| Description: | $C_{386}$ |
| Order: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Generators: |
$b^{193}, b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, 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: | $C_{386}:C_{64}$ |
| Order: | \(24704\)\(\medspace = 2^{7} \cdot 193 \) |
| Exponent: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| 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: | $C_{64}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(64\)\(\medspace = 2^{6} \) |
| Automorphism Group: | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_{192}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_{64}$, of order \(64\)\(\medspace = 2^{6} \) |
Related subgroups
| Centralizer: | $C_{386}$ | |
| Normalizer: | $C_{386}:C_{64}$ | |
| Complements: | $C_{64}$ $C_{64}$ | |
| Minimal over-subgroups: | $D_{386}$ | |
| Maximal under-subgroups: | $C_{193}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{193}:C_{64}$ |