Subgroup ($H$) information
| Description: | $C_3\times C_{96}$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Generators: |
$a^{6}, a^{48}, a^{24}, a^{64}, a^{96}, b^{386}, a^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_6\times F_{193}$ |
| Order: | \(222336\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 193 \) |
| 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{579}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_8\times \GL(2,3)$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $193$ |
| Möbius function | $-2$ |
| Projective image | $C_2\times F_{193}$ |