Subgroup ($H$) information
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(193\) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, b^{193}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{386}:C_4$ |
| Order: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Exponent: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_4$ | ||
| Normalizer: | $C_2\times C_4$ | ||
| Normal closure: | $C_{386}:C_4$ | ||
| Core: | $C_2^2$ | ||
| Minimal over-subgroups: | $C_{386}:C_4$ | ||
| Maximal under-subgroups: | $C_2^2$ | $C_4$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $193$ |
| Möbius function | $-1$ |
| Projective image | $D_{193}$ |