Subgroup ($H$) information
| Description: | $C_{119}$ |
| Order: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Index: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Exponent: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Generators: |
$b^{20315}, b^{1673}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{28441}:C_{34}$ |
| Order: | \(966994\)\(\medspace = 2 \cdot 7 \cdot 17^{2} \cdot 239 \) |
| Exponent: | \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 17$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{239}:C_{34}$ |
| Order: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Exponent: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Automorphism Group: | $F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| Outer Automorphisms: | $C_{14}$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 17$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4063}.C_{357}.C_8.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{28441}:C_{34}$ | ||
| Normalizer: | $C_{28441}:C_{34}$ | ||
| Complements: | $C_{239}:C_{34}$ | ||
| Minimal over-subgroups: | $C_{28441}$ | $C_{17}\times C_{119}$ | $C_{238}$ |
| Maximal under-subgroups: | $C_{17}$ | $C_7$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-239$ |
| Projective image | $C_{239}:C_{34}$ |