Subgroup ($H$) information
| Description: | $C_{18528}.C_4$ |
| Order: | \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(18528\)\(\medspace = 2^{5} \cdot 3 \cdot 193 \) |
| Generators: |
$b^{192}, a^{96}, b^{1158}, a^{48}, b^{12352}, b^{18528}, b^{2316}, b^{4632}, b^{9264}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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: | $C_2\times C_{48}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $D_4:C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $D_4:C_2^3$, of order \(64\)\(\medspace = 2^{6} \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(455344128\)\(\medspace = 2^{18} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_{386}.C_{12}.C_8^2.C_2^4$ |
| $W$ | $F_{193}$, 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 conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2\times F_{193}$ |