Subgroup ($H$) information
| Description: | $C_{193}$ |
| Order: | \(193\) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(193\) |
| Generators: |
$b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $193$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{193}:C_{64}$ |
| Order: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Exponent: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.\OD_{16}$, of order \(1185792\)\(\medspace = 2^{11} \cdot 3 \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_{192}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{6176}$ |
| Normalizer: | $C_{193}:C_{64}$ |
| Complements: | $C_{64}$ |
| Minimal over-subgroups: | $C_{386}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{193}:C_{64}$ |