Subgroup ($H$) information
| Description: | $C_{238}$ |
| Order: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Index: | \(239\) |
| Exponent: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Generators: |
$b^{1673}, b^{478}, a$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
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 $p = 17$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{239}:(C_2\times C_{714})$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{238}$ | ||
| Normalizer: | $C_{238}$ | ||
| Normal closure: | $C_{1673}:C_{34}$ | ||
| Core: | $C_{14}$ | ||
| Minimal over-subgroups: | $C_{1673}:C_{34}$ | ||
| Maximal under-subgroups: | $C_{119}$ | $C_{34}$ | $C_{14}$ |
Other information
| Number of subgroups in this conjugacy class | $239$ |
| Möbius function | $-1$ |
| Projective image | $C_{239}:C_{17}$ |