Subgroup ($H$) information
| Description: | $C_{119}:C_{68}$ |
| Order: | \(8092\)\(\medspace = 2^{2} \cdot 7 \cdot 17^{2} \) |
| Index: | \(7\) |
| Exponent: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Generators: |
$b^{85}, b^{7}, a^{238}, a^{28}, a^{119}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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: | $C_7$ |
| Order: | \(7\) |
| Exponent: | \(7\) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{119}.C_6^2.C_8^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{119}.C_6.C_8^2.C_2^3$ |
| $W$ | $D_{119}$, of order \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_7\times D_{119}$ |