Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| Exponent: | \(3\) |
| Generators: |
$a^{16}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{193}:C_{48}$ |
| Order: | \(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \) |
| Exponent: | \(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{48}$ | |
| Normalizer: | $C_{48}$ | |
| Normal closure: | $C_{193}:C_3$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_{193}:C_3$ | $C_6$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $193$ |
| Möbius function | $0$ |
| Projective image | $C_{193}:C_{48}$ |