Subgroup ($H$) information
| Description: | $C_{1544}:C_{24}$ |
| Order: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Generators: |
$b^{772}, b^{16}, b^{1544}, a^{24}, a^{12}, a^{6}, b^{386}, a^{16}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{3088}:C_{48}$ |
| Order: | \(148224\)\(\medspace = 2^{8} \cdot 3 \cdot 193 \) |
| Exponent: | \(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1544}.C_{96}.C_2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{772}.C_{96}.C_2^4$ |
| $W$ | $C_{193}:C_{48}$, of order \(9264\)\(\medspace = 2^{4} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{16}$ | |||
| Normalizer: | $C_{3088}:C_{48}$ | |||
| Minimal over-subgroups: | $C_{3088}:C_{24}$ | $C_{1544}:C_{48}$ | ||
| Maximal under-subgroups: | $C_{1544}:C_{12}$ | $C_{772}:C_{24}$ | $C_{1544}:C_8$ | $C_8\times C_{24}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $C_{386}:C_{48}$ |