Subgroup ($H$) information
| Description: | $C_{3346}$ |
| Order: | \(3346\)\(\medspace = 2 \cdot 7 \cdot 239 \) |
| Index: | \(17\) |
| Exponent: | \(3346\)\(\medspace = 2 \cdot 7 \cdot 239 \) |
| Generators: |
$b^{1673}, b^{478}, b^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, maximal, a semidirect factor, cyclic (hence abelian, elementary ($p = 2,7,239$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{1673}:C_{34}$ |
| Order: | \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| Exponent: | \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 17$.
Quotient group ($Q$) structure
| Description: | $C_{17}$ |
| Order: | \(17\) |
| Exponent: | \(17\) |
| Automorphism Group: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{239}:(C_2\times C_{714})$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{714}$, of order \(1428\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 17 \) |
| $W$ | $C_{17}$, of order \(17\) |
Related subgroups
| Centralizer: | $C_{3346}$ | ||
| Normalizer: | $C_{1673}:C_{34}$ | ||
| Complements: | $C_{17}$ | ||
| Minimal over-subgroups: | $C_{1673}:C_{34}$ | ||
| Maximal under-subgroups: | $C_{1673}$ | $C_{478}$ | $C_{14}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{239}:C_{17}$ |