Subgroup ($H$) information
| Description: | $A_4^2.S_4\wr C_2.C_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(3,7,11)(4,8,12)(19,20)(23,24), (3,12,7)(4,11,8)(15,24,20)(16,23,19)(17,18) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $A_4^2.S_4^2:C_2^3.D_4$ |
| Order: | \(5308416\)\(\medspace = 2^{16} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_3^4.C_2.C_2^5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^8.C_3^4.C_2^5.C_2^3$ |
| $W$ | $A_4^2:\POPlus(4,3).C_2^4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^2\wr C_2.C_2^2.C_2^4$ |
| Normal closure: | $A_4^2\wr C_2.C_2^2.C_2^3$ |
| Core: | $C_2^9.C_3^4:D_4$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |