Subgroup ($H$) information
| Description: | $C_{386}$ |
| Order: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(386\)\(\medspace = 2 \cdot 193 \) |
| Generators: |
$a^{12}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{1544}:C_3$ |
| Order: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Quotient group ($Q$) structure
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{193}.C_{96}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{192}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{192}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(772\)\(\medspace = 2^{2} \cdot 193 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_{1544}$ | |
| Normalizer: | $C_{1544}:C_3$ | |
| Minimal over-subgroups: | $C_{193}:C_6$ | $C_{772}$ |
| Maximal under-subgroups: | $C_{193}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{193}:C_{12}$ |