Subgroup ($H$) information
| Description: | $C_2^6.D_6^2$ |
| Order: | \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,36)(5,8)(6,9)(14,20)(23,25)(33,34), (10,26)(13,28)(15,29)(19,31), (3,20) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.D_6^2$ |
| Order: | \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_6^2.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_5^4:D_4:C_2$, of order \(221184\)\(\medspace = 2^{13} \cdot 3^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6.D_6^2$ |
| Normal closure: | $C_2^8.D_6^2$ |
| Core: | $C_2^4:D_4\times D_6$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |