Subgroup ($H$) information
| Description: | $C_{193}:C_{96}$ |
| Order: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Generators: |
$a^{24}b^{2676}, a^{48}b^{1728}, b^{2316}, a^{12}b^{3510}, b^{1544}, a^{6}b^{3483}, b^{24}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{4632}.C_{96}$ |
| Order: | \(444672\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 193 \) |
| 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.
Quotient group ($Q$) structure
| Description: | $C_{24}$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{2316}.C_{96}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{193}.C_{96}.C_2^3$ |
| $W$ | $C_{193}:C_{96}$, of order \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{24}$ | ||
| Normalizer: | $C_{4632}.C_{96}$ | ||
| Minimal over-subgroups: | $C_{579}:C_{96}$ | $D_{193}:C_{96}$ | |
| Maximal under-subgroups: | $C_{193}:C_{48}$ | $C_{193}:C_{32}$ | $C_{96}$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_{772}:C_{96}$ |