Subgroup ($H$) information
| Description: | $C_{17}\times C_{34}$ |
| Order: | \(578\)\(\medspace = 2 \cdot 17^{2} \) |
| Index: | \(239\) |
| Exponent: | \(34\)\(\medspace = 2 \cdot 17 \) |
| Generators: |
$b^{4063}, a, b^{478}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, elementary for $p = 17$ (hence hyperelementary), and metacyclic.
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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4063}.C_{476}.C_2^2.C_2$ |
| $\operatorname{Aut}(H)$ | $\GL(2,17)$, of order \(78336\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 17 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $239$ |
| Möbius function | $-1$ |
| Projective image | $C_{239}:C_{17}$ |