Subgroup ($H$) information
| Description: | $C_2^{12}.S_4^3:C_2$ |
| Order: | \(113246208\)\(\medspace = 2^{22} \cdot 3^{3} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(23,24)(35,36), (5,6)(7,13,8,14)(9,16,10,15)(11,18,12,17)(25,26)(27,28) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{10}.(S_4^2.S_4\wr C_2)$ |
| Order: | \(679477248\)\(\medspace = 2^{23} \cdot 3^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(21743271936\)\(\medspace = 2^{28} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | Group of order \(1811939328\)\(\medspace = 2^{26} \cdot 3^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | not computed |
| Projective image | not computed |