Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(28441\)\(\medspace = 7 \cdot 17 \cdot 239 \) |
| Exponent: | \(2\) |
| Generators: |
$b^{239}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), a $2$-Sylow subgroup (hence a Hall subgroup), a $p$-group, simple, and rational.
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: | $C_{239}:C_{119}$ |
| Order: | \(28441\)\(\medspace = 7 \cdot 17 \cdot 239 \) |
| Exponent: | \(28441\)\(\medspace = 7 \cdot 17 \cdot 239 \) |
| Automorphism Group: | $F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{239}:C_{238}$ | ||
| Normalizer: | $C_{239}:C_{238}$ | ||
| Complements: | $C_{239}:C_{119}$ | ||
| Minimal over-subgroups: | $C_{478}$ | $C_{34}$ | $C_{14}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $-239$ |
| Projective image | $C_{239}:C_{119}$ |