Subgroup ($H$) information
| Description: | $C_7^3:D_6$ |
| Order: | \(4116\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{3} \) |
| Index: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(15,17,19,21,16,18,20)(22,24,26,28,23,25,27), (22,28,27,26,25,24,23), (8,10,12,14,9,11,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $D_7\wr S_4$ |
| Order: | \(921984\)\(\medspace = 2^{7} \cdot 3 \cdot 7^{4} \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^4.C_2^3.(C_6\times S_4)$, of order \(2765952\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{4} \) |
| $\operatorname{Aut}(H)$ | $C_7^2.C_3^3.C_2^4$ |
| $W$ | $C_7^2:D_6$, of order \(588\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $112$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $D_7\wr S_4$ |