Subgroup ($H$) information
| Description: | $C_{1158}:C_{96}$ |
| Order: | \(111168\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 193 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Generators: |
$b^{96}, b^{6176}, b^{9264}, a^{6}b^{14403}, a^{64}, a^{96}b^{17040}, a^{12}b^{16710}, a^{24}b^{8268}, a^{48}b^{8568}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{18528}.C_{192}$ |
| Order: | \(3557376\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 193 \) |
| 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: | $C_{32}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Automorphism Group: | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(113836032\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_{579}.C_{96}.C_2^3$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |