Subgroup ($H$) information
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(249\)\(\medspace = 3 \cdot 83 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a$
|
| 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 $2$-Sylow subgroup (hence a Hall subgroup), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_3:C_{664}$ |
| Order: | \(1992\)\(\medspace = 2^{3} \cdot 3 \cdot 83 \) |
| Exponent: | \(1992\)\(\medspace = 2^{3} \cdot 3 \cdot 83 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{246}:C_2^3$, of order \(1968\)\(\medspace = 2^{4} \cdot 3 \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(S)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{664}$ | |
| Normalizer: | $C_{664}$ | |
| Normal closure: | $C_3:C_8$ | |
| Core: | $C_4$ | |
| Minimal over-subgroups: | $C_{664}$ | $C_3:C_8$ |
| Maximal under-subgroups: | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $S_3\times C_{83}$ |