Subgroup ($H$) information
| Description: | $C_{1158}:C_{96}$ |
| Order: | \(111168\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 193 \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Generators: |
$b^{192}, b^{12352}, b^{18528}, a^{18}b^{24}, a^{64}, a^{96}b^{32832}, a^{36}b^{20208}, a^{72}b^{33120}, a^{144}b^{12096}$
|
| 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: | $C_{193}:C_{192}^2$ |
| Order: | \(7114752\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 193 \) |
| 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: | $C_2\times C_{32}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Automorphism Group: | $C_8.C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_8.C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 \(455344128\)\(\medspace = 2^{18} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_{579}.C_{96}.C_2^3$ |
| $W$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{192}$ |
| Normalizer: | $C_{193}:C_{192}^2$ |
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | not computed |
| Projective image | not computed |