Subgroup ($H$) information
| Description: | $C_7:C_4$ |
| Order: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Index: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a^{119}, b, a^{238}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_7:C_{476}$ |
| Order: | \(3332\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 17 \) |
| 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: | $C_{119}$ |
| Order: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Exponent: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Automorphism Group: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 7,17$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{48}\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{238}$ | |
| Normalizer: | $C_7:C_{476}$ | |
| Complements: | $C_{119}$ $C_{119}$ $C_{119}$ $C_{119}$ | |
| Minimal over-subgroups: | $C_7:C_{68}$ | $C_7:C_{28}$ |
| Maximal under-subgroups: | $C_{14}$ | $C_4$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_7\times C_{119}$ |