Subgroup ($H$) information
| Description: | $C_7:C_6$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$b^{3}, c^{22}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $D_{11}\times F_7$ |
| Order: | \(924\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 11 \) |
| Exponent: | \(462\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_7\times F_{11}$, of order \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_2\times F_7$ | ||
| Normal closure: | $C_{77}:C_6$ | ||
| Core: | $C_7:C_3$ | ||
| Minimal over-subgroups: | $C_{77}:C_6$ | $C_2\times F_7$ | |
| Maximal under-subgroups: | $C_7:C_3$ | $C_{14}$ | $C_6$ |
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $1$ |
| Projective image | $D_{11}\times F_7$ |