Subgroup ($H$) information
| Description: | $C_{193}$ |
| Order: | \(193\) |
| Index: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(193\) |
| Generators: |
$b^{16}$
|
| 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: | $C_{3088}.C_{96}$ |
| Order: | \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_8\times C_{192}$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Automorphism Group: | $C_2.C_4^3.C_2^6.C_2$ |
| Outer Automorphisms: | $C_2.C_4^3.C_2^6.C_2$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1544}.C_{96}.C_2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{192}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_{96}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{3088}$ | |||
| Normalizer: | $C_{3088}.C_{96}$ | |||
| Complements: | $C_8\times C_{192}$ | |||
| Minimal over-subgroups: | $C_{193}:C_3$ | $C_{386}$ | $D_{193}$ | $D_{193}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{3088}.C_{96}$ |