Subgroup ($H$) information
| Description: | $C_{193}:C_4$ |
| Order: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Index: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Generators: |
$a^{48}b^{664}, a^{96}b^{5040}, b^{32}$
|
| 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: | $C_{18528}:C_{64}$ |
| Order: | \(1185792\)\(\medspace = 2^{11} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_8\times C_{192}$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Automorphism Group: | $C_2.C_4^3.C_2^6.C_2$ |
| Outer Automorphisms: | $C_2.C_4^3.C_2^6.C_2$ |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{3088}.C_{24}.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_{64}$, of order \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{96}$ | ||
| Normalizer: | $C_{18528}:C_{64}$ | ||
| Minimal over-subgroups: | $C_{193}:C_{12}$ | $C_{386}:C_4$ | $C_{193}:C_8$ |
| Maximal under-subgroups: | $D_{193}$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_{18528}:C_{64}$ |