Subgroup ($H$) information
| Description: | $C_{1544}.C_8$ |
| Order: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
| Generators: |
$b^{1544}, a^{8}b^{5932}, b^{3088}, a^{4}b^{4742}, b^{3860}, a^{2}b^{4643}, b^{32}$
|
| 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: | $C_{6176}.C_{16}$ |
| Order: | \(98816\)\(\medspace = 2^{9} \cdot 193 \) |
| Exponent: | \(12352\)\(\medspace = 2^{6} \cdot 193 \) |
| 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: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1544}.C_{24}.C_8^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{772}.C_{96}.C_2^4$ |
| $W$ | $C_{193}:C_{16}$, of order \(3088\)\(\medspace = 2^{4} \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{32}$ | |||
| Normalizer: | $C_{6176}.C_{16}$ | |||
| Minimal over-subgroups: | $C_{3088}:C_8$ | |||
| Maximal under-subgroups: | $C_{1544}:C_4$ | $D_{193}:C_{16}$ | $D_{193}:C_{16}$ | $C_4\times C_{16}$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{772}:C_{16}$ |