Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(3474\)\(\medspace = 2 \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(2\) |
| Generators: |
$a^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_{2316}:C_3$ |
| 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), hyperelementary for $p = 3$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{1158}:C_3$ |
| Order: | \(3474\)\(\medspace = 2 \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(1158\)\(\medspace = 2 \cdot 3 \cdot 193 \) |
| Automorphism Group: | $C_{579}.C_{192}.C_2$ |
| Outer Automorphisms: | $S_3\times C_{64}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:(C_2\times \OD_{16})$, of order \(444672\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{2316}:C_3$ | |||||
| Normalizer: | $C_{2316}:C_3$ | |||||
| Minimal over-subgroups: | $C_{386}$ | $C_6$ | $C_6$ | $C_6$ | $C_6$ | $C_4$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $579$ |
| Projective image | $C_{1158}:C_3$ |