Subgroup ($H$) information
| Description: | $C_{772}:C_{48}$ |
| Order: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \) |
| Generators: |
$b^{386}, b^{4}, a^{96}, a^{48}, a^{24}, a^{12}, b^{193}, a^{64}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_4\times F_{193}$ |
| Order: | \(148224\)\(\medspace = 2^{8} \cdot 3 \cdot 193 \) |
| 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: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{386}.C_{96}.C_2^3$ |
| $W$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_4$ | ||||
| Normalizer: | $C_4\times F_{193}$ | ||||
| Minimal over-subgroups: | $C_{772}:C_{96}$ | ||||
| Maximal under-subgroups: | $C_{772}:C_{24}$ | $C_{386}:C_{48}$ | $C_{386}:C_{48}$ | $C_{772}:C_{16}$ | $C_4\times C_{48}$ |
Other information
| Möbius function | $0$ |
| Projective image | $F_{193}$ |