Subgroup ($H$) information
| Description: | $C_2^6.D_6^2$ |
| Order: | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| Index: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,5)(3,4)(8,15,10,12)(9,13,14,11), (8,9,10,14)(11,12,13,15), (1,4)(3,5) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $S_7\times C_2\wr D_4$ |
| Order: | \(645120\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_2^4.C_2^2.A_7.C_2$ |
| $\operatorname{Aut}(H)$ | $C_5^4:D_4:C_2$, of order \(221184\)\(\medspace = 2^{13} \cdot 3^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $D_4^2.(D_6\times S_4)$ |
| Normal closure: | $S_7\times C_2\wr C_2^2$ |
| Core: | $C_2\wr C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $70$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |