Subgroup ($H$) information
| Description: | not computed |
| Order: | \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(11,12)(13,14)(15,16)(17,18)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34), (13,14) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and metabelian (hence solvable). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2.C_2\wr D_6$ |
| Order: | \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^8.C_3^3.C_2^6$ |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^4:C_3.C_2^4.C_6.C_2^3$ |
| Normal closure: | $C_2^4:C_3.A_4^2.C_2^3$ |
| Core: | $A_4^2.C_2^6$ |
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 | $C_2^8.(C_6\times S_3^2)$ |