Subgroup ($H$) information
| Description: | $C_{239}:C_{34}$ |
| Order: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Index: | \(17\) |
| Exponent: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| Generators: |
$b^{4063}, ab^{12}, b^{34}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct 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: | $C_{4063}:C_{34}$ |
| Order: | \(138142\)\(\medspace = 2 \cdot 17^{2} \cdot 239 \) |
| Exponent: | \(8126\)\(\medspace = 2 \cdot 17 \cdot 239 \) |
| 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: | $C_{17}$ |
| Order: | \(17\) |
| Exponent: | \(17\) |
| Automorphism Group: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4063}.C_{476}.C_2^2.C_2$ |
| $\operatorname{Aut}(H)$ | $F_{239}$, of order \(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \) |
| $W$ | $C_{239}:C_{17}$, of order \(4063\)\(\medspace = 17 \cdot 239 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{4063}:C_{17}$ |