Subgroup ($H$) information
| Description: | $C_{193}:C_4$ |
| Order: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| Generators: |
$a^{6}b^{1155}, b^{1158}, b^{12}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{2316}:C_{12}$ |
| Order: | \(27792\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_3\times C_{12}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times \GL(2,3)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times \GL(2,3)$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1158}.C_{96}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_{12}$, of order \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $C_{2316}:C_{12}$ | ||
| Complements: | $C_3\times C_{12}$ | ||
| Minimal over-subgroups: | $C_{193}:C_{12}$ | $C_{193}:C_{12}$ | $C_4\times D_{193}$ |
| Maximal under-subgroups: | $C_{386}$ | $C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{1158}:C_{12}$ |