Subgroup ($H$) information
| Description: | $C_{193}:C_4$ |
| Order: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Generators: |
$a^{2}b^{2}, b^{772}, b^{8}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{1544}:C_4$ |
| Order: | \(6176\)\(\medspace = 2^{5} \cdot 193 \) |
| Exponent: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_4$, of order \(772\)\(\medspace = 2^{2} \cdot 193 \) |
Related subgroups
| Centralizer: | $C_8$ | ||
| Normalizer: | $C_{1544}:C_4$ | ||
| Minimal over-subgroups: | $C_4\times D_{193}$ | $C_{193}:C_8$ | $C_{193}:C_8$ |
| Maximal under-subgroups: | $C_{386}$ | $C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{772}:C_4$ |