Subgroup ($H$) information
| Description: | $C_{239}$ |
| Order: | \(239\) |
| Index: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Exponent: | \(239\) |
| Generators: |
$b^{14}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $239$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
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: | $C_{238}$ |
| Order: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Exponent: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Automorphism Group: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{239}:(C_2\times C_{714})$ |
| $\operatorname{Aut}(H)$ | $C_{238}$, of order \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| $W$ | $C_{17}$, of order \(17\) |
Related subgroups
| Centralizer: | $C_{3346}$ | ||
| Normalizer: | $C_{1673}:C_{34}$ | ||
| Complements: | $C_{238}$ | ||
| Minimal over-subgroups: | $C_{239}:C_{17}$ | $C_{1673}$ | $C_{478}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{1673}:C_{34}$ |