Subgroup ($H$) information
| Description: | $C_2\times A_4^3:S_4$ |
| Order: | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(3,11,7)(5,12,9)(17,20)(18,19), (3,11,7), (2,10,6,8)(3,7,11)(4,5)(13,15) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.S_3\wr S_3$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^3.C_2^6:S_4$, of order \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $S_4^3.D_6$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| $W$ | $A_4^3.(C_2\times S_4)$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^7.S_3\wr S_3$ |
| Normal closure: | $A_4^3.C_2^3:S_4$ |
| Core: | $A_4^3.C_2^3$ |
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $S_4^3.S_4$ |