Subgroup ($H$) information
| Description: | $C_2\times F_7$ |
| Order: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$ab^{129}, b^{88}, c, b^{66}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $D_{154}:C_6$ |
| Order: | \(1848\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 11 \) |
| Exponent: | \(924\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{77}.C_{30}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $D_{14}:C_6$ | |||
| Normal closure: | $C_{154}:C_6$ | |||
| Core: | $C_7:C_6$ | |||
| Minimal over-subgroups: | $C_{154}:C_6$ | $D_{14}:C_6$ | ||
| Maximal under-subgroups: | $C_7:C_6$ | $F_7$ | $D_{14}$ | $C_2\times C_6$ |
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $1$ |
| Projective image | $D_{11}\times F_7$ |