Subgroup ($H$) information
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{6}b^{1155}, b^{772}, b^{1158}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{2316}:C_{12}$ |
| Order: | \(27792\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1158}.C_{96}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{12}^2$ | ||
| Normalizer: | $C_{12}^2$ | ||
| Normal closure: | $C_{193}:C_{12}$ | ||
| Core: | $C_6$ | ||
| Minimal over-subgroups: | $C_{193}:C_{12}$ | $C_3\times C_{12}$ | $C_2\times C_{12}$ |
| Maximal under-subgroups: | $C_6$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $193$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{386}:C_{12}$ |