Subgroup ($H$) information
| Description: | $C_7\times D_7$ |
| Order: | \(98\)\(\medspace = 2 \cdot 7^{2} \) |
| Index: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$ab^{3}c^{5}d^{9}, d^{3}, c$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $\He_7:C_6^2$ |
| Order: | \(12348\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_7.C_3^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6\times F_7$, of order \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $3$ |
| Projective image | $\He_7:C_6^2$ |