Subgroup ($H$) information
| Description: | $C_2^5.S_4\wr C_2$ |
| Order: | \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| Index: | \(3\) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(2,4)(6,7)(10,12)(13,15), (9,11)(10,12), (9,10)(11,12), (1,3)(2,4)(9,12) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_2^2:S_4)^2.D_6$ |
| Order: | \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2:S_4)^2.D_6$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2^6.S_4\wr C_2$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \) |
| $W$ | $C_2^5.S_4\wr C_2$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_2^5.S_4\wr C_2$ |
| Normal closure: | $(C_2^2:S_4)^2.D_6$ |
| Core: | $C_2^8.\SOPlus(4,2)$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(C_2^2:S_4)^2.D_6$ |