Subgroup ($H$) information
| Description: | $C_{1158}:C_{96}$ |
| Order: | \(111168\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 193 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Generators: |
$b^{24}, b^{1544}, b^{2316}, a^{18}b^{363}, a^{64}, a^{96}b^{2088}, a^{36}b^{2598}, a^{72}b^{1932}, a^{144}b^{2232}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{24}\times F_{193}$ |
| Order: | \(889344\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 193 \) |
| 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: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{2316}.C_{96}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{579}.C_{96}.C_2^3$ |
| $W$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_4\times F_{193}$ |