Subgroup ($H$) information
| Description: | $C_2^{12}.S_4^3:C_2$ |
| Order: | \(113246208\)\(\medspace = 2^{22} \cdot 3^{3} \) |
| Index: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,20,8)(2,19,7)(3,24,10,6,22,11)(4,23,9,5,21,12)(13,14)(15,17)(16,18)(25,31,38,26,32,37) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6.(C_2^{12}.(D_6\times S_7))$ |
| Order: | \(15854469120\)\(\medspace = 2^{24} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(31708938240\)\(\medspace = 2^{25} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | 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 | $70$ |
| Möbius function | not computed |
| Projective image | not computed |