Subgroup ($H$) information
| Description: | $C_{119}$ |
| Order: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Index: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Generators: |
$b^{85}, b^{7}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{119}:C_{476}$ |
| Order: | \(56644\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17^{2} \) |
| Exponent: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{476}$ |
| Order: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Automorphism Group: | $C_2^2\times C_{48}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2\times C_{48}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{119}.C_6^2.C_8^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{119}\times C_{238}$ | ||
| Normalizer: | $C_{119}:C_{476}$ | ||
| Complements: | $C_{476}$ | ||
| Minimal over-subgroups: | $C_{17}\times C_{119}$ | $C_7\times C_{119}$ | $C_{238}$ |
| Maximal under-subgroups: | $C_{17}$ | $C_7$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{119}:C_{476}$ |