Subgroup ($H$) information
| Description: | $C_{239}:C_{34}$ |
| Order: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Index: | \(833\)\(\medspace = 7^{2} \cdot 17 \) |
| Exponent: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Generators: |
$a^{119}, a^{14}b^{63}, b^{119}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 17$.
Ambient group ($G$) information
| Description: | $C_{28441}:C_{238}$ |
| Order: | \(6768958\)\(\medspace = 2 \cdot 7^{2} \cdot 17^{2} \cdot 239 \) |
| 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), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_7\times C_{119}$ |
| Order: | \(833\)\(\medspace = 7^{2} \cdot 17 \) |
| Exponent: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Automorphism Group: | $C_{16}\times C_6.\SO(3,7)$ |
| Outer Automorphisms: | $C_{16}\times C_6.\SO(3,7)$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 7$ (hence hyperelementary), and metacyclic.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{28441}.C_{357}.C_8.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{239}$, 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
Other information
| Number of subgroups in this autjugacy class | $17$ |
| Number of conjugacy classes in this autjugacy class | $17$ |
| Möbius function | $-7$ |
| Projective image | $C_{28441}:C_{119}$ |