Subgroup ($H$) information
| Description: | $C_4\times D_{193}$ |
| Order: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Generators: |
$a^{8}b^{3076}, b^{13896}, b^{96}, b^{9264}$
|
| 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.
Ambient group ($G$) information
| Description: | $C_{18528}.C_{16}$ |
| Order: | \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
| 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: | $C_4\times C_{48}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| Outer Automorphisms: | $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1544}.C_{24}.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{193}.C_{96}.C_2^3$ |
| $W$ | $C_{193}:C_{16}$, of order \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | not computed |