Subgroup ($H$) information
| Description: | $C_{239}:C_{34}$ |
| Order: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Index: | \(7\) |
| Exponent: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Generators: |
$b^{239}, a^{7}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 17$.
Ambient group ($G$) information
| Description: | $C_{239}:C_{238}$ |
| 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 and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_7$ |
| Order: | \(7\) |
| Exponent: | \(7\) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| $\operatorname{Aut}(H)$ | $F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| $W$ | $C_{239}:C_{119}$, of order \(28441\)\(\medspace = 7 \cdot 17 \cdot 239 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_{239}:C_{238}$ | ||
| Complements: | $C_7$ | ||
| Minimal over-subgroups: | $C_{239}:C_{238}$ | ||
| Maximal under-subgroups: | $C_{239}:C_{17}$ | $C_{478}$ | $C_{34}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{239}:C_{119}$ |