Subgroup ($H$) information
| Description: | $C_{193}:C_6$ |
| Order: | \(1158\)\(\medspace = 2 \cdot 3 \cdot 193 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(1158\)\(\medspace = 2 \cdot 3 \cdot 193 \) |
| Generators: |
$a^{6}, a^{4}b^{2}, b^{3}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{579}:C_{12}$ |
| Order: | \(6948\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 193 \) |
| 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: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{579}.C_{192}.C_2$ |
| $\operatorname{Aut}(H)$ | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| $W$ | $C_{193}:C_{12}$, of order \(2316\)\(\medspace = 2^{2} \cdot 3 \cdot 193 \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $C_{579}:C_{12}$ | ||
| Minimal over-subgroups: | $C_{579}:C_6$ | $C_{193}:C_{12}$ | |
| Maximal under-subgroups: | $C_{193}:C_3$ | $D_{193}$ | $C_6$ |
| Autjugate subgroups: | 6948.a.6.b1.b1 | 6948.a.6.b1.c1 |
Other information
| Möbius function | $1$ |
| Projective image | $C_{579}:C_{12}$ |