Subgroup ($H$) information
| Description: | $C_{193}:C_8$ |
| Order: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Generators: |
$a^{2}b^{2}, a^{4}b^{20}, b^{4}, a^{8}b^{96}$
|
| Derived length: | $2$ |
The subgroup is 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: | $C_{772}:C_{16}$ |
| Order: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Exponent: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{386}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_{16}$, of order \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
Related subgroups
| Centralizer: | $C_4$ | ||
| Normalizer: | $C_{772}:C_{16}$ | ||
| Minimal over-subgroups: | $C_{386}:C_8$ | $C_{193}:C_{16}$ | $C_{193}:C_{16}$ |
| Maximal under-subgroups: | $C_{193}:C_4$ | $C_8$ | |
| Autjugate subgroups: | 12352.1674.8.e1.b1 |
Other information
| Möbius function | $0$ |
| Projective image | $C_{772}:C_{16}$ |