Subgroup ($H$) information
| Description: | $A_4:S_4^2$ |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(13,15)(14,16), (5,8)(6,7)(9,11)(10,12)(13,16)(14,15), (2,4)(5,7,8,6)(11,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^8.C_3^4:\OD_{16}$ |
| Order: | \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_4^2\wr C_2.C_2^2.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $S_4^3:C_2$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \) |
| $W$ | $A_4:S_4^2$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $A_4^3.C_2^3$ |
| Normal closure: | $C_2^8.C_3^3.D_6$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^8.C_3^4:\OD_{16}$ |